# Is there any point to the 12 times table?

June 27, 2013 7:18 AM Subscribe

Because 13 is unlucky and 14 is just stupid. Duh.

posted by phunniemee at 7:21 AM on June 27, 2013 [1 favorite]

posted by phunniemee at 7:21 AM on June 27, 2013 [1 favorite]

11 is the most fun on the times table.

posted by shakespeherian at 7:24 AM on June 27, 2013 [19 favorites]

posted by shakespeherian at 7:24 AM on June 27, 2013 [19 favorites]

The pre-decimal system in Britain was not actually that mad if you did a lot of mental calculation. Figures like 240 (the number of pence in a pound) and 12 (pence in a shilling) divide into lots of different whole numbers.

It's the same reason why we don't have decimalised time, or decimalised degrees in a circle.

posted by Bare Ruined Choirs at 7:25 AM on June 27, 2013 [10 favorites]

It's the same reason why we don't have decimalised time, or decimalised degrees in a circle.

posted by Bare Ruined Choirs at 7:25 AM on June 27, 2013 [10 favorites]

*It's the same reason why we don't have decimalised time, or decimalised degrees in a circle.*

I use a decimilized week. It's currently sexday.

posted by goethean at 7:27 AM on June 27, 2013 [30 favorites]

It involves the old english standard of the dozen. How many dozens of donuts do I need to feed twenty people if each person eats three? How many dozens of eggs do I need if I need to bake thirty cakes that need 3 eggs each?

Also, 10 is neat - but it presents a conceptual wall. By leaning 11 and 12 times, you learn the pattern of how larger numbers multiply without resorting to pen and paper every time you hit two columns.

posted by Slap*Happy at 7:28 AM on June 27, 2013 [30 favorites]

Also, 10 is neat - but it presents a conceptual wall. By leaning 11 and 12 times, you learn the pattern of how larger numbers multiply without resorting to pen and paper every time you hit two columns.

posted by Slap*Happy at 7:28 AM on June 27, 2013 [30 favorites]

Do vocational schools for bakers teach the 13 times table?

posted by schmod at 7:29 AM on June 27, 2013 [12 favorites]

posted by schmod at 7:29 AM on June 27, 2013 [12 favorites]

I don't know why anyone would want to multiply 12 times 12. It's just gross.

posted by Longtime Listener at 7:29 AM on June 27, 2013 [50 favorites]

posted by Longtime Listener at 7:29 AM on June 27, 2013 [50 favorites]

The 12 times table is gross.

posted by Quonab at 7:29 AM on June 27, 2013 [5 favorites]

posted by Quonab at 7:29 AM on June 27, 2013 [5 favorites]

Slap*Happy's got it.

Also, it's handy for converting feet to inches.

posted by maryr at 7:30 AM on June 27, 2013 [2 favorites]

Also, it's handy for converting feet to inches.

posted by maryr at 7:30 AM on June 27, 2013 [2 favorites]

*decimalised degrees in a circle*

Not heard of the gradian? It's what the grad button on calculators is for.

Admittedly it is about as popular for angles as Rankine is for temperature.

posted by edd at 7:30 AM on June 27, 2013 [5 favorites]

You know, he mentions the money thing in the very first paragraph of the article. He's wondering if there is any point

posted by absalom at 7:30 AM on June 27, 2013

*today*.posted by absalom at 7:30 AM on June 27, 2013

I would think it is helpful because many things are sold in dozens, especially food products. So that makes learning my 12x tables something I learned in school that I actually use in life.

Mmmm donuts.

posted by NoraCharles at 7:30 AM on June 27, 2013

Mmmm donuts.

posted by NoraCharles at 7:30 AM on June 27, 2013

12 is much better than 10 as a base for a number system. 12 can be divided by 2, 3, 4, and 6. There are approximately 12 lunar months in the year. It's no accident that many cultures have math that revolves around the number 12, and that we have a word for twelve items (a dozen) but no common word ten items.

If only humans had 6 fingers on each hand we'd be all set.

posted by alms at 7:33 AM on June 27, 2013 [10 favorites]

If only humans had 6 fingers on each hand we'd be all set.

posted by alms at 7:33 AM on June 27, 2013 [10 favorites]

*Also, 10 is neat - but it presents a conceptual wall. By leaning 11 and 12 times, you learn the pattern of how larger numbers multiply without resorting to pen and paper every time you hit two columns.*

Came in here to say exactly this. In fact, I personally would advocate going back up to the 20 times table like they used to do.

posted by DU at 7:33 AM on June 27, 2013 [3 favorites]

12 is better than 10 for a LOT of operations. Having roots of by 2, 3, 4 and 6 help show all sorts of the ways numbers interconnect to a young mind.

Slap*Happy's response is spot on too.

PS: Rote memorization of the multiplication tables without explaining HOW it works has always been wrong. If by the year after a kid can't extend it out to 13,14 and 15 as quickly as he can write the teacher(s) failed that student.

posted by DigDoug at 7:34 AM on June 27, 2013

Slap*Happy's response is spot on too.

PS: Rote memorization of the multiplication tables without explaining HOW it works has always been wrong. If by the year after a kid can't extend it out to 13,14 and 15 as quickly as he can write the teacher(s) failed that student.

posted by DigDoug at 7:34 AM on June 27, 2013

*It's easier to do "times 6" and then double the result.*

Times SIX? What a waste, just do "times 3" and double the result.

All we really need are the prime number times tables, and there's fewer of those anyway!

posted by grog at 7:34 AM on June 27, 2013 [6 favorites]

Schoolhouse Rock 12 times song.

Bonus: my favorite haunting 8 times table song.

posted by emjaybee at 7:35 AM on June 27, 2013 [5 favorites]

Bonus: my favorite haunting 8 times table song.

posted by emjaybee at 7:35 AM on June 27, 2013 [5 favorites]

12 is a dozen. Dozen is probably still a common unit in various areas.

posted by Thorzdad at 7:35 AM on June 27, 2013

posted by Thorzdad at 7:35 AM on June 27, 2013

I'm a little disappointed the answer isn't "no, cuz you can just use a calculator if you have to do timeses", because I pretty much was on to that when I was 7 or 8. However, I also determined back then it is stupid that I have to show my work, so because I got the right answer I should still get full marks.

posted by Hoopo at 7:36 AM on June 27, 2013 [1 favorite]

posted by Hoopo at 7:36 AM on June 27, 2013 [1 favorite]

Don't forget the number of inches in a foot for us crazy people still using imperial measurements.

posted by cmfletcher at 7:38 AM on June 27, 2013 [1 favorite]

posted by cmfletcher at 7:38 AM on June 27, 2013 [1 favorite]

Also, Time. Sexagesimal and 12 hour blocks... Time is way harder than metric systems. And will stay that way until we change the length of a second!

posted by DigDoug at 7:38 AM on June 27, 2013

posted by DigDoug at 7:38 AM on June 27, 2013

I just convert everything to base 12 and do the math that way, it comes out much cleaner.

1x1012=1012, 2x1012=2012, 3x1012=3012,...

posted by backseatpilot at 7:39 AM on June 27, 2013

1x1012=1012, 2x1012=2012, 3x1012=3012,...

posted by backseatpilot at 7:39 AM on June 27, 2013

And, Jesus, yes. We need more education that focuses less on rote memorization and computation. I'm very glad that there are intelligent people out there who are debating and thinking about this.

I specifically remember being quizzed in middle school on "estimation." I once had points docked from a test because my answers were too accurate. It's helpful to know good estimation strategies (am I seriously the only person who can calculate a ~18% tip in my head?), but the ways that they were taught always seemed a bit crazy to me. Similarly, I think that you can make a really great argument for teaching the times tables up to 10, but not beyond that.

Even at that, rote memorization of multiplication tables seems fairly useless. As long as you know a few multiplication shortcuts, you can fill in the table with a very small amount of effort (I

That said, I maintain that metric units are great for science, and shitty for everyday life. Human-scale comparisons are usually imprecise and awkward.

posted by schmod at 7:40 AM on June 27, 2013 [2 favorites]

I specifically remember being quizzed in middle school on "estimation." I once had points docked from a test because my answers were too accurate. It's helpful to know good estimation strategies (am I seriously the only person who can calculate a ~18% tip in my head?), but the ways that they were taught always seemed a bit crazy to me. Similarly, I think that you can make a really great argument for teaching the times tables up to 10, but not beyond that.

Even at that, rote memorization of multiplication tables seems fairly useless. As long as you know a few multiplication shortcuts, you can fill in the table with a very small amount of effort (I

*still*get tripped up on 7s for some reason). If you're doing enough math, the values will eventually begin to stick. Start out by multiplying by 6, then two, and if you're doing that often enough, you'll remember it. In school, I forgot the 12 times table almost immediately after being quizzed on it, because it was so rarely used. However, now that I deal with multiples of twelve often (for whatever reason), I can quickly rattle off the values, because I use them often in my daily life.That said, I maintain that metric units are great for science, and shitty for everyday life. Human-scale comparisons are usually imprecise and awkward.

posted by schmod at 7:40 AM on June 27, 2013 [2 favorites]

*In fact, I personally would advocate going back up to the 20 times table like they used to do.*

Had to memorize uptil the 20 times table for my mother. School only asked until the 12 times table fwiw and that's what's stayed in the decades since.

posted by infini at 7:41 AM on June 27, 2013

*If only humans had 6 fingers on each hand we'd be all set.*

We actually have twelve digits on each hand.

posted by Slap*Happy at 7:42 AM on June 27, 2013 [8 favorites]

*am I seriously the only person who can calculate a ~18% tip in my head?*

17.5% would be a lot easier. It's 10% + 5% + 2.5%. So on a $45 bill, say, it's $4.5 + $2.25 + $1.12 ~= $8.90. Confirmed by 20% (or 1/5) of 45 being 9.

posted by DU at 7:42 AM on June 27, 2013 [1 favorite]

*We actually have twelve digits on each hand.*

Add the two thumb segments and the front and back of the hand itself and you've got yourself a really useful base.

posted by DU at 7:43 AM on June 27, 2013

123FOURFIVE678NINETEN11Twe-e-e-e-e-lve!

posted by jimmythefish at 7:43 AM on June 27, 2013 [7 favorites]

posted by jimmythefish at 7:43 AM on June 27, 2013 [7 favorites]

18% is 10% times 2, then subtract 10% of the result.

posted by decathecting at 7:45 AM on June 27, 2013 [5 favorites]

posted by decathecting at 7:45 AM on June 27, 2013 [5 favorites]

Bakers have to learn to multiply up to 13.

posted by twoleftfeet at 7:46 AM on June 27, 2013 [1 favorite]

posted by twoleftfeet at 7:46 AM on June 27, 2013 [1 favorite]

DU: "

I usually go with (10%*2) - (1%*2)

It boils down to two decimal shifts, one very easy multiplication (by 2), and a fairly easy subtraction.

Or, if you're drunk, 20% minus a bit.

posted by schmod at 7:46 AM on June 27, 2013

*17.5% would be a lot easier. It's 10% + 5% + 2.5%. So on a $45 bill, say, it's $4.5 + $2.25 + $1.12 ~= $8.90. Confirmed by 20% (or 1/5) of 45 being 9.*"I usually go with (10%*2) - (1%*2)

It boils down to two decimal shifts, one very easy multiplication (by 2), and a fairly easy subtraction.

Or, if you're drunk, 20% minus a bit.

posted by schmod at 7:46 AM on June 27, 2013

%n: "

They told me it was to prove you weren't cheating. I got out of that one by grabbing another random quiz sheet and filling out the answers while there was nobody but me and the teacher in the room. I asked if he still didn't think I could do long division in my head. He let it go.

posted by Karmakaze at 7:46 AM on June 27, 2013 [1 favorite]

*I'm a little disappointed the answer isn't "no, cuz you can just use a calculator if you have to do timeses", because I pretty much was on to that when I was 7 or 8. However, I also determined back then it is stupid that I have to show my work, so because I got the right answer I should still get full marks.*"They told me it was to prove you weren't cheating. I got out of that one by grabbing another random quiz sheet and filling out the answers while there was nobody but me and the teacher in the room. I asked if he still didn't think I could do long division in my head. He let it go.

posted by Karmakaze at 7:46 AM on June 27, 2013 [1 favorite]

I usually just give a 20% tip. Or round up 5-10 bucks depending on the tab. Easier math, more money for the waitress, win-win.

posted by emjaybee at 7:47 AM on June 27, 2013 [2 favorites]

posted by emjaybee at 7:47 AM on June 27, 2013 [2 favorites]

Diverging offtopic, why the hell is Britain still using the stone as a unit of weight? That shit is

posted by schmod at 7:47 AM on June 27, 2013 [24 favorites]

*base-14*. Ain't nobody got time for that.posted by schmod at 7:47 AM on June 27, 2013 [24 favorites]

*18% is 10% times 2, then subtract 10% of the result.*

That's how I convert form Celsius to Fahrenheit.

posted by Obscure Reference at 7:49 AM on June 27, 2013 [2 favorites]

Brits are stoned, clearly.

posted by alms at 7:50 AM on June 27, 2013 [1 favorite]

posted by alms at 7:50 AM on June 27, 2013 [1 favorite]

I can't calculate anything in my head.

When I was in therapy the doctor determined the root of my crippling math anxiety is childhood trauma over memorizing the multiplication tables.

Fuck times 12, times 12 fucking sucks.

That was cathartic.

posted by Ad hominem at 7:50 AM on June 27, 2013 [1 favorite]

When I was in therapy the doctor determined the root of my crippling math anxiety is childhood trauma over memorizing the multiplication tables.

Fuck times 12, times 12 fucking sucks.

That was cathartic.

posted by Ad hominem at 7:50 AM on June 27, 2013 [1 favorite]

*I usually just give a 20% tip. Or round up 5-10 bucks depending on the tab. Easier math, more money for the waitress, win-win.*

I knew a guy who'd always add "a dollar for tax" because .85 was too hard. There is an idiotic grace to that. Wildly overpay for small items, and wildly, wildly underpay for large items. it would be extremely rare for the dollar to be the exact correct amount, but it all sorts itself out over the long run, right?

posted by dirtdirt at 7:51 AM on June 27, 2013

*I can't calculate anything in my head.*

It's really super easy once you practice a bit. And it gets easier to add new tricks as you notice patterns. My math-oriented officemate the other day was a little bamboozled by me multiplying 73 * 3 in my head.

Me: It was easy. 70 * 3 + 3 * 3.

Her: Well, I didn't know 70 * 3.

Me: *blankly looking at her* What's 7 * 3?

Her: Oh.

posted by DU at 7:55 AM on June 27, 2013 [1 favorite]

*That's how I convert form Celsius to Fahrenheit.*

I always thought that was just double plus 30.

*However, I also determined back then it is stupid that I have to show my work, so because I got the right answer I should still get full marks.*

I had to argue with a calculus professor because I was doing integrals in my head. Once I solved a few on the board instantly without writing anything down he relented and gave me the marks. Still couldn't explain how I did it I just ended up writing down the correct answer. The easiest way I could explain it was that I just look at it and imagine what I would need to derivate to get the thing I was trying to integrate.

posted by koolkat at 7:56 AM on June 27, 2013 [3 favorites]

Have you ever built anything out of dimensional lumber?

posted by unSane at 7:56 AM on June 27, 2013 [2 favorites]

posted by unSane at 7:56 AM on June 27, 2013 [2 favorites]

Sexagesimal (base 60) is an interesting case. We use it to measure time but we don't have 60 symbols we just 'map' it into base ten and divide the digits with colons. So 12:45 is really 'symbol12symbol45'.

The only base that we regularly use more than ten symbols is hexadecimal where we use 0123456789ABCDEF.

I think the way of 'mapping' one base to another could be extended. For example we could map 36 into binary. 36->11:110

Or perhaps time into hexadecimal 12:45->C:2D

I'd really like to make a program that did this automatically.

posted by Confess, Fletch at 7:56 AM on June 27, 2013 [3 favorites]

The only base that we regularly use more than ten symbols is hexadecimal where we use 0123456789ABCDEF.

I think the way of 'mapping' one base to another could be extended. For example we could map 36 into binary. 36->11:110

Or perhaps time into hexadecimal 12:45->C:2D

I'd really like to make a program that did this automatically.

posted by Confess, Fletch at 7:56 AM on June 27, 2013 [3 favorites]

*I got the right answer I should still get full marks.*

They told me it was to prove you weren't cheating.

They told me it was to prove you weren't cheating.

They told you wrong.

The reasons you need to show your work are:

a) When you get the wrong answer, you have something physical to do forensics on (e.g. forgot to carry the one vs have no clue what you are doing).

b) When you have a harder problem you will need to write it down (everyone will eventually reach this level, but the level will differ by person). Getting into the habit of writing it down means you get over this hump rather than thinking you've run into a wall.

posted by DU at 8:00 AM on June 27, 2013 [22 favorites]

Most people seem to have some version of "break down weird/unusual/large problem into smaller/easier problems, add/subtract, get result." It's interesting that so many people do it in different ways.

posted by emjaybee at 8:02 AM on June 27, 2013

posted by emjaybee at 8:02 AM on June 27, 2013

12-bar Blues. You don't need to count, just wait for the drummer.

posted by mule98J at 8:05 AM on June 27, 2013 [5 favorites]

posted by mule98J at 8:05 AM on June 27, 2013 [5 favorites]

*It's easier to do "times 6" and then double the result.*

...

Times SIX? What a waste, just do "times 3" and double the result.

...

Times SIX? What a waste, just do "times 3" and double the result.

That's ridiculous. To multiply something by 12, just do "times .48" and then "times 25" the result.

posted by Greg Nog at 8:05 AM on June 27, 2013 [9 favorites]

I think it's an interesting mental calisthenic. You don't have to do them, but it's a base to build upon. When you start seeing interesting patterns, I think there's a tendency to explore them. If you're discouraged from seeing them, you'll never get curious.

Confess, Fletch: "

You're making somebody at Swatch very happy right now.

posted by boo_radley at 8:06 AM on June 27, 2013 [2 favorites]

Confess, Fletch: "

*Or perhaps time into hexadecimal 12:45->C:2D*"You're making somebody at Swatch very happy right now.

posted by boo_radley at 8:06 AM on June 27, 2013 [2 favorites]

Yo Mamma so shortsighted, she doesn't realize you can't have "The Dozens" without the 12 times table.

posted by radwolf76 at 8:10 AM on June 27, 2013 [2 favorites]

posted by radwolf76 at 8:10 AM on June 27, 2013 [2 favorites]

One divided by seven is 0.14285714285...

Two divided by seven is 0.28571428571...

Three divided by seven is 0.42857142857...

Four divided by seven is 0.57142857142...

Five divided by seven is 0.71428571428...

So you can see the pattern, right? It's the same digits rolled around in a cycle.

Remember 142857 and just roll the digits around. Multiply 1/7 = .142857... by 2 to get .285714... Multiply by 3 to get .428571

Just multiply the first two decimal places... if you know 142857 you can figure out where to start to start the cycle.

What's 53 divided by seven? OK,... seven times seven is 49, with four left over. So your answer is 7 with a decimal point that begins with 4 times 14, which is 56, so you start the cycle at 5. That is, 53/7 = 7.57142857142857142857...

posted by twoleftfeet at 8:11 AM on June 27, 2013 [19 favorites]

Two divided by seven is 0.28571428571...

Three divided by seven is 0.42857142857...

Four divided by seven is 0.57142857142...

Five divided by seven is 0.71428571428...

So you can see the pattern, right? It's the same digits rolled around in a cycle.

Remember 142857 and just roll the digits around. Multiply 1/7 = .142857... by 2 to get .285714... Multiply by 3 to get .428571

Just multiply the first two decimal places... if you know 142857 you can figure out where to start to start the cycle.

What's 53 divided by seven? OK,... seven times seven is 49, with four left over. So your answer is 7 with a decimal point that begins with 4 times 14, which is 56, so you start the cycle at 5. That is, 53/7 = 7.57142857142857142857...

posted by twoleftfeet at 8:11 AM on June 27, 2013 [19 favorites]

Diverging offtopic, why the hell is Britain still using the stone as a unit of weight? That shit is base-14. Ain't nobody got time for that.It's only customary now for body weight, so there's not a whole lot of need for timesing and dividing. Even then, simple pounds or kg is becoming more common.

posted by Jehan at 8:17 AM on June 27, 2013 [1 favorite]

*That's how I convert form Celsius to Fahrenheit.*

That is extremely rude.

posted by Celsius1414 at 8:18 AM on June 27, 2013 [3 favorites]

As an architect who can never remember in a split-second how many feet and inches are in say 76" or 82" or 98", I wish I had memorized the 12 times table as a child. Even after 10 years, I still have to pause to go 60, 72, 84... OK 84" is an even foot... now how many feet is that? So minus 2"... that give me 6'-10".

It just takes me twice as long and occasionally make a mistake.

posted by yeti at 8:18 AM on June 27, 2013 [2 favorites]

It just takes me twice as long and occasionally make a mistake.

posted by yeti at 8:18 AM on June 27, 2013 [2 favorites]

twoleftfeet, that reminds me of the part in Concrete Mathematics where he's like "...in the lingo of computer programming, we get J(n) from n by doing a one-bit cyclic shift left! Magic ... [i]f we had been working all along in binary notation, we probably would have spotted this pattern immediately."

posted by 7segment at 8:19 AM on June 27, 2013 [1 favorite]

posted by 7segment at 8:19 AM on June 27, 2013 [1 favorite]

I was told there would be no math.

posted by briank at 8:20 AM on June 27, 2013 [11 favorites]

posted by briank at 8:20 AM on June 27, 2013 [11 favorites]

I just realized that as someone who works in the beverage industry, lots of my product is packaged in 6, 12, & 24 packs. So I kinda know my 12 tables. But that's the only time in my day when it comes in useful

And Bob Dorough's alien in Little Twelve-Toes didn't teach me my 12x at all. Love th groove on th song, but an educational failure.

posted by Pirate-Bartender-Zombie-Monkey at 8:21 AM on June 27, 2013

And Bob Dorough's alien in Little Twelve-Toes didn't teach me my 12x at all. Love th groove on th song, but an educational failure.

posted by Pirate-Bartender-Zombie-Monkey at 8:21 AM on June 27, 2013

%n: "

I was told to follow my happiness. Now I have no savings.

posted by boo_radley at 8:24 AM on June 27, 2013 [2 favorites]

*I was told there would be no math.*"I was told to follow my happiness. Now I have no savings.

posted by boo_radley at 8:24 AM on June 27, 2013 [2 favorites]

*that reminds me of the part in Concrete Mathematics...*

It's a generating function mod 2. Either that, or a modular function of genus 2. I can't remember which.

posted by twoleftfeet at 8:24 AM on June 27, 2013 [1 favorite]

I went to French schools and the metric system was the only system of measurement taught. Neanmoins, we memorized the 12 times table, and the 11 times, for that matter.

posted by mareli at 8:27 AM on June 27, 2013

posted by mareli at 8:27 AM on June 27, 2013

Inchworm song that gives you a multiplication earworm.

posted by emjaybee at 8:30 AM on June 27, 2013

posted by emjaybee at 8:30 AM on June 27, 2013

I suffered in my arithmetic classes because at my school we had to bring our books to class and also our own lunches. They would ask me to "carry the 1", but I couldn't carry it because my hands were full with my books and my lunch.

posted by twoleftfeet at 8:34 AM on June 27, 2013 [1 favorite]

posted by twoleftfeet at 8:34 AM on June 27, 2013 [1 favorite]

I find it easier to do arithmetic in base 12 if I use my fingers and my toes, and cut off eight of my toes.

posted by twoleftfeet at 8:38 AM on June 27, 2013 [1 favorite]

posted by twoleftfeet at 8:38 AM on June 27, 2013 [1 favorite]

I was taught how to do manual square roots once during an enrichment class, but can't remember enough of the algorithm to repeat it now. You had to know your squares up to 100 and it sort of resembled doing long division two digits at a time.

posted by ceribus peribus at 8:42 AM on June 27, 2013

posted by ceribus peribus at 8:42 AM on June 27, 2013

*All we really need are the prime number times tables, and there's fewer of those anyway*

Nope, the set of prime numbers is the same size as the set of all non-prime numbers. Or indeed the set of all numbers, prime or non-prime. Or so I've been told.

posted by Grangousier at 8:48 AM on June 27, 2013 [2 favorites]

Here's a quick trick. What's 65 squared? C'mon, what's 65 times 65?

Do this. Take the digits before the last 5. In this case, that means 6. Multiply by the number one greater; in this case, that means 7. Now, 6 times 7 is 42. Then just tack on a 25 at the end.

That is, 65 times 65 is 4225.

This always works with numbers that end in 5. Again, another example. What is 45 squared? Just do 4 times 5 and stick a 25 at the end. So 45

posted by twoleftfeet at 8:50 AM on June 27, 2013 [12 favorites]

Do this. Take the digits before the last 5. In this case, that means 6. Multiply by the number one greater; in this case, that means 7. Now, 6 times 7 is 42. Then just tack on a 25 at the end.

That is, 65 times 65 is 4225.

This always works with numbers that end in 5. Again, another example. What is 45 squared? Just do 4 times 5 and stick a 25 at the end. So 45

^{2}= 2025.posted by twoleftfeet at 8:50 AM on June 27, 2013 [12 favorites]

*Here's a quick trick. What's 65 squared? C'mon, what's 65 times 65?*

/hits command-space

/types "65^2" in Spotlight field

4225

posted by Celsius1414 at 8:52 AM on June 27, 2013 [5 favorites]

twoleftfeet, you are hurting my brain with all these magic tricks. Why couldn't any of my math teachers clue me into any magic tricks other than the 9 times one? I think I would've been inspired more with a greater understanding of these shenanigans to look for other interesting tricks.

posted by vuron at 8:54 AM on June 27, 2013

posted by vuron at 8:54 AM on June 27, 2013

*Do this. Take the digits before the last 5.*

Because I have a terrible memory for infrequently-used tricks, I prefer the more general trick to square any number.

`(a+b)`

65

^{2}= a^{2}+ b^{2}+ 2ab65

^{2}= (60 + 5)^{2}= 3600 + 25 + 2*60*5 = 3625 + 600 = 4225posted by DU at 8:55 AM on June 27, 2013 [4 favorites]

%n: "

I actually had been conscientiously showing all work for the first few days. By the time I copped the attitude I was deep into "I learned this two weeks ago, can we please have a new topic now?" territory. My school, in its infinite wisdom, had placed me in the wrong math section. That's also why I was so annoyed about being accused of copying off of other students. As if I needed to.

posted by Karmakaze at 9:02 AM on June 27, 2013

*They told you wrong.*

The reasons you need to show your work are:

a) When you get the wrong answer, you have something physical to do forensics on (e.g. forgot to carry the one vs have no clue what you are doing).

b) When you have a harder problem you will need to write it down (everyone will eventually reach this level, but the level will differ by person). Getting into the habit of writing it down means you get over this hump rather than thinking you've run into a wall."The reasons you need to show your work are:

a) When you get the wrong answer, you have something physical to do forensics on (e.g. forgot to carry the one vs have no clue what you are doing).

b) When you have a harder problem you will need to write it down (everyone will eventually reach this level, but the level will differ by person). Getting into the habit of writing it down means you get over this hump rather than thinking you've run into a wall.

I actually had been conscientiously showing all work for the first few days. By the time I copped the attitude I was deep into "I learned this two weeks ago, can we please have a new topic now?" territory. My school, in its infinite wisdom, had placed me in the wrong math section. That's also why I was so annoyed about being accused of copying off of other students. As if I needed to.

posted by Karmakaze at 9:02 AM on June 27, 2013

*(a+b)*

^{2}= a^{2}+ b^{2}+ 2abRight. That's why that particular trick works (the b

^{2}makes the 25 at the end, the 2ab pushes the next decimal up... think about it.)

The single best math trick I know of - not entirely a math trick mind you - is the Doomsday Algorithm for finding the day of the week given the date. (What day of the week is November 15, this year?)

You can learn the basic technique (for this year or the next, say) in a few minutes and it's both extremely impressive and very useful. I've taught this to six year olds and to people who thought they had no aptitude for math whatsoever.

Try it!

posted by twoleftfeet at 9:02 AM on June 27, 2013 [6 favorites]

Am I the only person here who was not, in fact, taught a 12x12 times table but rather only through the 10x10 times table? I can figure out my elevenzes and twelvezes right quick, but I initially did so by taking the Nx10 figure and adding N or 2N. The lower end of the 11s and 12s eventually became committed to memory (as did 12x12 because it's so gross), but yeah, there is a lightning speed calculation that goes on to arrive at something like 9x12.

Maybe this is because I learned multiplication in the midst of the Great Metric Panic of 1976.

posted by drlith at 9:08 AM on June 27, 2013 [1 favorite]

Maybe this is because I learned multiplication in the midst of the Great Metric Panic of 1976.

posted by drlith at 9:08 AM on June 27, 2013 [1 favorite]

Of course the 12 times table is useful. I was in a rug store yesterday and had to use the 12x table a bunch of times in my head to figure out the inch dimensions of various rugs so I'd know if they would work with my dining table. So yeah, as long as we have non-metric units where 12 is a grouping (inches/feet, dozens) people should know the 12x table.

posted by freecellwizard at 9:10 AM on June 27, 2013

posted by freecellwizard at 9:10 AM on June 27, 2013

*When you have a harder problem you will need to write it down (everyone will eventually reach this level, but the level will differ by person). Getting into the habit of writing it down means you get over this hump rather than thinking you've run into a wall.*

Most people, though, when they reach this point -- and they do, I agree -- will just start writing shit down if they aren't playing weird power games with the teacher about writing stuff down.

posted by jeather at 9:12 AM on June 27, 2013 [1 favorite]

I find the 18% calculation an overwrought habit of many Americans. Just do 20%. It doesn't require a calculator (anyone can reckon 10% of a number and double it, right?), and it's nicer to wait staff.

posted by Jubal Kessler at 9:12 AM on June 27, 2013 [2 favorites]

posted by Jubal Kessler at 9:12 AM on June 27, 2013 [2 favorites]

%n: "

Alternately, one can divide by 5.

posted by Karmakaze at 9:14 AM on June 27, 2013 [1 favorite]

*I find the 18% calculation an overwrought habit of many Americans. Just do 20%. It doesn't require a calculator (anyone can reckon 10% of a number and double it, right?), and it's nicer to wait staff.*"Alternately, one can divide by 5.

posted by Karmakaze at 9:14 AM on June 27, 2013 [1 favorite]

Sorry, but I'm not taking math advice from someone who can't even count their left feet.

posted by It's Raining Florence Henderson at 9:15 AM on June 27, 2013 [3 favorites]

posted by It's Raining Florence Henderson at 9:15 AM on June 27, 2013 [3 favorites]

Can't I just hire some guy in China to do my multiplication for me? I'm too busy using my law degree to file patents for obvious software applications.

posted by happyroach at 9:16 AM on June 27, 2013 [2 favorites]

posted by happyroach at 9:16 AM on June 27, 2013 [2 favorites]

I just patented the use of binary operations in microprocessing. Pay up!

posted by It's Raining Florence Henderson at 9:18 AM on June 27, 2013 [1 favorite]

posted by It's Raining Florence Henderson at 9:18 AM on June 27, 2013 [1 favorite]

%n: "

Occasionally, one has to split a bill with an auto-grat. Using 20% here usually results in several people overpaying, and one person getting a really cheap meal.

posted by schmod at 9:25 AM on June 27, 2013

*I find the 18% calculation an overwrought habit of many Americans. Just do 20%. It doesn't require a calculator (anyone can reckon 10% of a number and double it, right?), and it's nicer to wait staff.*"Occasionally, one has to split a bill with an auto-grat. Using 20% here usually results in several people overpaying, and one person getting a really cheap meal.

posted by schmod at 9:25 AM on June 27, 2013

I don;t know if it's dyspraxia or a quirk of my brain, but I was very bad at maths in school as I struggle with rote learning. (I know enough French vocab to have a stab at reading a magazine despite never having learned it at all, but ask me to do Spanish grammar and I'd have no idea - despite six years of lessons, the grammar just didn't stick. Which was a pain when I went on to do linguistics and could not for the life of me memorise the IPA.) I could do quadratic equations, because they made some kind of sense to me - it was like cracking a code - and I can remember a lot of stuff about averages because the practical applications made sense to me, but I don't even know the seven or eight times tables, and some others I have to work out by doing some kind of adding and dividing in my head. I've often bought furniture that's too large or too small, because I can't visualise measurements in real life - I end up saying 'but what

I was SO glad to be able to turn 16 and drop maths for good, as it was incredibly frustrating to a) not know why things were the way they were b) having the teacher angry with me for not knowing why things were the way they were, because she thought I was just lazy rather than finding it hard to remember and apply things.

posted by mippy at 9:25 AM on June 27, 2013 [1 favorite]

*size*is that', hoping for a comparison to a bicycle or a postbox or something, and people think they're smart for replying 'well, 16in is 16in' and smirking. (I can visualise weight by imagining butter or flour, and liquids by Coke bottles, so that's less difficult.) Yet ask me to match up one colour exactly with another, or remember a paragraph verbatim, and I can do it, so being stupid at maths is quite embarrassing.I was SO glad to be able to turn 16 and drop maths for good, as it was incredibly frustrating to a) not know why things were the way they were b) having the teacher angry with me for not knowing why things were the way they were, because she thought I was just lazy rather than finding it hard to remember and apply things.

posted by mippy at 9:25 AM on June 27, 2013 [1 favorite]

*I was very bad at maths in school as I struggle with rote learning.*

You might want to try learning math on your own, since there's nothing rote about any of it other than basic arithmetic.

posted by DU at 9:29 AM on June 27, 2013 [1 favorite]

As has been mentioned, the reason Brits used to do it was because of pre-decimal money, there being 12 pence to the shilling. I even asked my junior school teacher why we stopped at 12 and this was the very reason he gave me.

posted by Decani at 9:33 AM on June 27, 2013

posted by Decani at 9:33 AM on June 27, 2013

*I find the 18% calculation an overwrought habit of many Americans. Just do 20%. It doesn't require a calculator (anyone can reckon 10% of a number and double it, right?), and it's nicer to wait staff.*

I mentally calculate both 15% and 20% in my head, then select a number somewhere between those two. Or higher than the latter, if I have been drinking and am therefore feeling great love for the world.

posted by Greg Nog at 9:34 AM on June 27, 2013

I can't really explain it well, but anything that involves memorising formulas or tables - mathematical rules, verb declension, music theory calculation etc - does not seem to stick in my brain at all. Yet I seem to have an eidetic memory when it comes to written prose, song lyrics etc. 'Rote learning' is the best description I could think of of this type of information - I was always far far better at the interpretative subjects, so much so that my teachers for maths etc. thought I was lazy rather than merely a type of stupid.

posted by mippy at 9:34 AM on June 27, 2013

posted by mippy at 9:34 AM on June 27, 2013

I remember reading some years ago about a trick to squaring numbers near 50. Basically the pattern is:

For x ± 50: 50

I've actually managed to make use of that a couple of times.

posted by Emperor SnooKloze at 9:34 AM on June 27, 2013

For x ± 50: 50

^{2}- (50-x * 100)+ (50-x)^{2}so 49^{2}= 2500 -100 +1 = 2401; 47^{2}= 2209, and 51^{2}= 2500 -(-100) + 1 = 2601; 52^{2}= 2704, etc.I've actually managed to make use of that a couple of times.

posted by Emperor SnooKloze at 9:34 AM on June 27, 2013

*You might want to try learning math on your own, since there's nothing rote about any of it other than basic arithmetic.*

Are we including geometry in maths?

posted by Celsius1414 at 9:35 AM on June 27, 2013

In the U.S. and adjacent countries, 8.5" x 11" is the standard size for office paper (because of certain reasons.) Which is a bit irritating, considering that 12" is a standard unit (for other reasons.)

But if you take a standard sheet of office paper and fold one corner over to the opposite side, the diagonal will be almost exactly 12 inches (it's the square root of 8.5

So you can make a kind of ruler out of office paper.

posted by twoleftfeet at 9:37 AM on June 27, 2013 [8 favorites]

But if you take a standard sheet of office paper and fold one corner over to the opposite side, the diagonal will be almost exactly 12 inches (it's the square root of 8.5

^{2}+ 8.5^{2}, by Pythagoras.)So you can make a kind of ruler out of office paper.

posted by twoleftfeet at 9:37 AM on June 27, 2013 [8 favorites]

*That's how I convert form Celsius to Fahrenheit.*

*I always thought that was just double plus 30.*

Not a bad approximation. It's exactly right at 10 Celsius / 50 Fahrenheit. For every ten Celsius degrees you move away from that, you'll be off by two Fahrenheit degrees. So for example your method says 30 Celsius is 90 Fahrenheit; it's actually 86, which is off by 4.

Also, it inverts nicely; to convert from Fahrenheit to Celsius, take half minus 15.

posted by madcaptenor at 9:40 AM on June 27, 2013 [1 favorite]

*So you can see the pattern, right? It's the same digits rolled around in a cycle.*

Heh. I used this trick once to great effect during a stock negotiation at my company. Some divided-by-seven figure came up, the CEO pulls out his iPhone and calls out the integer quotient. Without missing a beat I call out the next six digits. CEO stares. I say, "yeah, engineers are good at math."

posted by ryanrs at 9:43 AM on June 27, 2013 [1 favorite]

twoleftfeet: "

The ISO paper sizes are *

posted by schmod at 9:44 AM on June 27, 2013

*But if you take a standard sheet of office paper and fold one corner over to the opposite side, the diagonal will be almost exactly 12 inches (it's the square root of 8.52 + 8.52).*"The ISO paper sizes are *

*SO** much more rational and easy to deal with. (Plus, A4's slightly-narrower aspect ratio makes it a bit easier to read single-column text than Letter-sized paper, which is really just a crime against typography in most places where it's used)posted by schmod at 9:44 AM on June 27, 2013

%n: "

After we run out of letters, we could always turn to Emoji or Unicode.

Mainly, because I really want to schedule a meeting at 4:CAT FACE WITH TEARS OF JOY.

posted by schmod at 9:46 AM on June 27, 2013 [5 favorites]

*Sexagesimal (base 60) is an interesting case. We use it to measure time but we don't have 60 symbols we just 'map' it into base ten and divide the digits with colons. So 12:45 is really 'symbol12symbol45*"After we run out of letters, we could always turn to Emoji or Unicode.

Mainly, because I really want to schedule a meeting at 4:CAT FACE WITH TEARS OF JOY.

posted by schmod at 9:46 AM on June 27, 2013 [5 favorites]

What is it about the number 12?

There are 12 inches in a foot, 12 months in a year, 12 signs of the zodiac, 12 disciples of Christ.

As they say, there are 12 steps in Alcoholics Anonymous and 12 ounces in a bottle of beer. Coincidence?

posted by twoleftfeet at 9:49 AM on June 27, 2013

There are 12 inches in a foot, 12 months in a year, 12 signs of the zodiac, 12 disciples of Christ.

As they say, there are 12 steps in Alcoholics Anonymous and 12 ounces in a bottle of beer. Coincidence?

posted by twoleftfeet at 9:49 AM on June 27, 2013

*Are we including geometry in maths?*

Sure. There's a lot of theorems, but you don't have to memorize them. Once you understand them and especially once you use them a few times, they are just basic facts that you know because they are obviously true without having to remember. You don''t need to drill them.

posted by DU at 9:49 AM on June 27, 2013

%n: "

I might not be understanding what you are getting at, but the first one is in use a bunch for hobbyist electronics (or basically anything with a 7-segment display). it's called Binary-Coded Decimal.

posted by ArgentCorvid at 9:50 AM on June 27, 2013

*Sexagesimal (base 60) is an interesting case. We use it to measure time but we don't have 60 symbols we just 'map' it into base ten and divide the digits with colons. So 12:45 is really 'symbol12symbol45'.*

The only base that we regularly use more than ten symbols is hexadecimal where we use 0123456789ABCDEF.

I think the way of 'mapping' one base to another could be extended. For example we could map 36 into binary. 36->11:110

Or perhaps time into hexadecimal 12:45->C:2D

I'd really like to make a program that did this automatically"The only base that we regularly use more than ten symbols is hexadecimal where we use 0123456789ABCDEF.

I think the way of 'mapping' one base to another could be extended. For example we could map 36 into binary. 36->11:110

Or perhaps time into hexadecimal 12:45->C:2D

I'd really like to make a program that did this automatically

I might not be understanding what you are getting at, but the first one is in use a bunch for hobbyist electronics (or basically anything with a 7-segment display). it's called Binary-Coded Decimal.

posted by ArgentCorvid at 9:50 AM on June 27, 2013

The first half of that article was great, but it lost me once it went into math I didn't learn in school. I think I get the gist of it though.

The 12 times table definitely made sense in the pre-decimal days for monetary calculations, which were probably the most common mental arithmetic you would do. But the pre decimal was (is) mad, because its not base 12. Sixteen ounces in a pound (lb), Fourteen pounds in a stone. Even if you get rid of stones, you still have to deal with 16 oz in a lb, and the US is still using that.

But I do agree that up to 12 is a good idea just for coping with larger numbers. 10 and 11 are too easy to do, whereas 12 requires a bit more thought.

posted by Joh at 9:52 AM on June 27, 2013 [1 favorite]

The 12 times table definitely made sense in the pre-decimal days for monetary calculations, which were probably the most common mental arithmetic you would do. But the pre decimal was (is) mad, because its not base 12. Sixteen ounces in a pound (lb), Fourteen pounds in a stone. Even if you get rid of stones, you still have to deal with 16 oz in a lb, and the US is still using that.

But I do agree that up to 12 is a good idea just for coping with larger numbers. 10 and 11 are too easy to do, whereas 12 requires a bit more thought.

posted by Joh at 9:52 AM on June 27, 2013 [1 favorite]

The ISO paper sizes are *SO* much more rational and easy to deal with.

"rational", heh.

"rational", heh.

The international paper size standard, ISO 216, is based on the German DIN 476 standard for paper sizes. ISO paper sizes are all based on a single aspect ratio ofposted by benito.strauss at 9:55 AM on June 27, 2013 [4 favorites]square root of 2, or approximately 1:1.4142.

*Sexagesimal (base 60) is an interesting case. We use it to measure time but we don't have 60 symbols we just 'map' it into base ten*

This is only because we have forgotten how to count to 12 using one hand, while using the other hand to do this five times.

posted by twoleftfeet at 10:08 AM on June 27, 2013

Here in Canada we like to say we're all metric and everything, but we're not. Height and weight of people is still measured in feet/inches and pounds. The 12 times table comes in handy for those feet and inches. Not to mention months and years. Of course, we don't use ounces (unless we are drug dealers), so I honestly have no idea how pounds break down.

But anyway, the twelve times table is

posted by Sys Rq at 10:08 AM on June 27, 2013

But anyway, the twelve times table is

*easy*. Not seeing what the big deal is.posted by Sys Rq at 10:08 AM on June 27, 2013

*Of course, we don't use ounces (unless we are drug dealers), so I honestly have no idea how pounds break down.*

So... you don't deal pounds?

posted by twoleftfeet at 10:10 AM on June 27, 2013

Confess, Fletch, ArgentCorvid: Binary-coded decimal was actually used as a native number format fairly frequently in the past (some processors, like Intels and others of that vintage, still have special instructions for doing arithmetic on numbers stored as BCD). I assume this is because doing number-base conversions for input and output is actually relatively expensive (you have to do a multiplication or division for each digit).

posted by hattifattener at 10:10 AM on June 27, 2013

posted by hattifattener at 10:10 AM on June 27, 2013

*Of course, we don't use ounces (unless we are drug dealers)*

In the States, the long-running joke is that drug dealers are probably the most knowledgable about the metric system outside scientists. :D

posted by Celsius1414 at 10:11 AM on June 27, 2013

You want

posted by zengargoyle at 10:26 AM on June 27, 2013

**TWELVES**!! I'll give you Twelve Tones (awesome new Vihart)posted by zengargoyle at 10:26 AM on June 27, 2013

I think the reason that the number 12 is important is that it is colossally abundant, unlike, say, 13 or 14.

This is probably also why we measure 360 degrees in a full circle instead of, say, 361 or 362.

posted by twoleftfeet at 10:29 AM on June 27, 2013

This is probably also why we measure 360 degrees in a full circle instead of, say, 361 or 362.

posted by twoleftfeet at 10:29 AM on June 27, 2013

ArgentCorvid: You're absolutely correct, another place we use one base 'mapped' into another.

posted by Confess, Fletch at 10:38 AM on June 27, 2013

posted by Confess, Fletch at 10:38 AM on June 27, 2013

ARGH YOU PEOPLE WITH YOUR CELSIUS APPROXIMATIONS

It is *NOT HARD AT ALL* to go from one to the other. Fahrenheit temp, minus 32, divided by 9, times 5. Celcius temp, divided by 5, times 9, plus 32.

(((F - 32) / 9) * 5) = C

(((C / 5) *9) +32) = F

50F? ((50-32 = 18) / 9 = 2) * 5 = 10C

25C? ((25 / 5 = 5) * 9 = 45) + 32 = 77F

212F? ((212 - 32 = 180) / 9 = 20) * 5 = 100C

-40F? ((-40 - 32 = -72) / 9 = -8) * 5 = -40C

-40C? ((-40 / 5 = -8) * 9 = -72) + 32 = -40F

posted by hanov3r at 10:43 AM on June 27, 2013 [6 favorites]

It is *NOT HARD AT ALL* to go from one to the other. Fahrenheit temp, minus 32, divided by 9, times 5. Celcius temp, divided by 5, times 9, plus 32.

(((F - 32) / 9) * 5) = C

(((C / 5) *9) +32) = F

50F? ((50-32 = 18) / 9 = 2) * 5 = 10C

25C? ((25 / 5 = 5) * 9 = 45) + 32 = 77F

212F? ((212 - 32 = 180) / 9 = 20) * 5 = 100C

-40F? ((-40 - 32 = -72) / 9 = -8) * 5 = -40C

-40C? ((-40 / 5 = -8) * 9 = -72) + 32 = -40F

posted by hanov3r at 10:43 AM on June 27, 2013 [6 favorites]

*Had to memorize uptil the 20 times table for my mother. School only asked until the 12 times table fwiw and that's what's stayed in the decades since.*

My mother was appalled that we only had to do up to 10. It's probably on her list entitled Things Wrong with the United States. That said, she wasn't so appalled as to make flash cards to make me learn up to 12. There's the inevitable tension between Americans Doing Things Wrong and We Should Do Things to Try and Fit In.

posted by hoyland at 10:45 AM on June 27, 2013 [2 favorites]

1414C ((1414 / 5 = 282.8) * 9 = 2545.2) + 32 = 2577.2.

IMO Fahrenheit2577.2 is not as catchy. ;)

posted by Celsius1414 at 10:46 AM on June 27, 2013 [1 favorite]

IMO Fahrenheit2577.2 is not as catchy. ;)

posted by Celsius1414 at 10:46 AM on June 27, 2013 [1 favorite]

How I convert Fahrenheit to Celsius in my head.

posted by madcaptenor at 10:48 AM on June 27, 2013 [1 favorite]

posted by madcaptenor at 10:48 AM on June 27, 2013 [1 favorite]

Anyone who thinks the twelve times table is redundant obvious hasn't work in retail. Everything comes twelve to a case, except cigarettes and Kellogs cereal.

posted by Static Vagabond at 10:59 AM on June 27, 2013

posted by Static Vagabond at 10:59 AM on June 27, 2013

You really have to learn all the way up to 20 x 20 if you want to have any chance of getting your name onto the high scores list on Number Munchers at your school. Other than that, what's the point?

As far as the whole "there is a word for dozen but not for ten of something," in French, although words exist for dozen (douzaine), the far more common is the word for "ten of something", which is dizaine. Eggs, though, do not come by the dizaine, instead they generally come in packs of 6 as neither a dozen nor a dizaine of eggs would fit in a French refrigerator anyway.

posted by artichoke_enthusiast at 11:03 AM on June 27, 2013 [2 favorites]

As far as the whole "there is a word for dozen but not for ten of something," in French, although words exist for dozen (douzaine), the far more common is the word for "ten of something", which is dizaine. Eggs, though, do not come by the dizaine, instead they generally come in packs of 6 as neither a dozen nor a dizaine of eggs would fit in a French refrigerator anyway.

posted by artichoke_enthusiast at 11:03 AM on June 27, 2013 [2 favorites]

I can convert -40 degrees between Celsius and Fahrenheit in less than 8.26719577 × 10 fortnights.

posted by twoleftfeet at 11:04 AM on June 27, 2013

^{-10}

posted by twoleftfeet at 11:04 AM on June 27, 2013

How else would I calculate how many boxes of eggs I need for daily 144 egg breakfast omelette.

posted by Hairy Lobster at 11:19 AM on June 27, 2013

posted by Hairy Lobster at 11:19 AM on June 27, 2013

Oh yeah. Like no American on this forum is thinking "we'll what about the doughnuts? Who will think of the Krispy Kreme?"

posted by hal_c_on at 11:23 AM on June 27, 2013

posted by hal_c_on at 11:23 AM on June 27, 2013

*It's easier to do "times 6" and then double the result.*

Easier yet to do "times 10" and add a couple. 17 x 12 is 170 plus 34, or 204

posted by weapons-grade pandemonium at 11:35 AM on June 27, 2013 [1 favorite]

*"What's 53 divided by seven? OK,... seven times seven is 49, with four left over. So your answer is 7 with a decimal point that begins with 4 times 14, which is 56, so you start the cycle at 5. That is, 53/7 = 7.57142857142857142857..."*

I just know that would make sense if only I understood it....

"If by the year after a kid can't extend it out to 13,14 and 15 as quickly as he can write the teacher(s) failed that student."

"If by the year after a kid can't extend it out to 13,14 and 15 as quickly as he can write the teacher(s) failed that student."

Nods intelligently - "Uh-huh. Yeah! Right on!!" *thinks* Wow, how could they even do that?

I've noticed before that people who get numbers really can't see what it is the mathslexic are missing. Incomprehension, she mutual.

posted by glasseyes at 11:38 AM on June 27, 2013

"Take the digits before the last 5. In this case, that means 6. Multiply by the number one greater; in this case, that means 7. Now, 6 times 7 is 42. Then just tack on a 25 at the end.

That is, 65 times 65 is 4225."

Uh-huh. Wow!

posted by glasseyes at 11:42 AM on June 27, 2013 [1 favorite]

That is, 65 times 65 is 4225."

Uh-huh. Wow!

posted by glasseyes at 11:42 AM on June 27, 2013 [1 favorite]

*If only humans had 6 fingers on each hand we'd be all set.*

THANKS ALOT, OBUMMER

posted by robocop is bleeding at 11:50 AM on June 27, 2013

*Sure. There's a lot of theorems, but you don't have to memorize them. Once you understand them and especially once you use them a few times, they are just basic facts that you know because they are obviously true without having to remember. You don''t need to drill them.*

Could you please travel back in time and tell my geometry teacher this? Thanks.

posted by emjaybee at 12:01 PM on June 27, 2013 [2 favorites]

Hairy Lobster: "

Get it together, stu.

posted by boo_radley at 12:01 PM on June 27, 2013

*How else would I calculate how many boxes of eggs I need for daily 144 egg breakfast omelette.*"Get it together, stu.

posted by boo_radley at 12:01 PM on June 27, 2013

twoleftfeet:

Once you have the sequence (which can be remembered as 7, 14, 28, 5—out of space, loop), you don't even have to multiply. For N/7ths, just start with the Nth smallest number. So 3/7ths should start with the third smallest number in 714285 which is 4, so 0.428571 repeating.

posted by fleacircus at 12:02 PM on June 27, 2013 [2 favorites]

*Just multiply the first two decimal places... if you know 142857 you can figure out where to start to start the cycle.*Once you have the sequence (which can be remembered as 7, 14, 28, 5—out of space, loop), you don't even have to multiply. For N/7ths, just start with the Nth smallest number. So 3/7ths should start with the third smallest number in 714285 which is 4, so 0.428571 repeating.

posted by fleacircus at 12:02 PM on June 27, 2013 [2 favorites]

I like the feel of certain numbers. I like the look of certain numbers. I like a whole bunch of numbers that turn up in the 7, 8, and 9 times tables: 42, 56, 49, 54, 45, 48, 63, 64. But where exactly they turn up in those tables I have never been able to fix in my mind since infancy.

I love the shape and feel of what happens when you multiply a number by itself. I get a little frisson of satisfaction thinking of 64, 49, 36 etc, like I just heard a rhyming couplet. But I have the times tables open now or I could never get the exact...i guess - hard to know how to put it - I wouldn't be able to marry up the correct notation with the exact value. I think the notation frisks about for me like an unreliable narrator...I cannot pin those pesky little squiggles down.

But when I was getting like 5% and 10% in arithmetic and algebra for end-of-year exams, I was top of the class in Geometry.

posted by glasseyes at 12:05 PM on June 27, 2013 [5 favorites]

I love the shape and feel of what happens when you multiply a number by itself. I get a little frisson of satisfaction thinking of 64, 49, 36 etc, like I just heard a rhyming couplet. But I have the times tables open now or I could never get the exact...i guess - hard to know how to put it - I wouldn't be able to marry up the correct notation with the exact value. I think the notation frisks about for me like an unreliable narrator...I cannot pin those pesky little squiggles down.

**mippy**, did you have trouble learning how to read an analogue clock face? I did, I think I was about 12 before I could. And I have to stop and collect myself in order to understand where is left and where is right. And funnily enough, I too was the first person in my class to understand quadratic equations, which gave the teacher a bit of a shock, I can tell you.But when I was getting like 5% and 10% in arithmetic and algebra for end-of-year exams, I was top of the class in Geometry.

posted by glasseyes at 12:05 PM on June 27, 2013 [5 favorites]

*did you have trouble learning how to read an analogue clock face? I did, I think I was about 12 before I could.*

If it's any consolation, I'm good at math, but as a kid I thought for longer than I like to admit that a "quarter hour" was 25 minutes because a quarter was 25 cents.

posted by Celsius1414 at 12:10 PM on June 27, 2013 [1 favorite]

*What's 65 squared?*

As a developer, I know my powers of 2. So it's gonna be a little higher than 64

^{2}= 4096.

posted by zompist at 12:10 PM on June 27, 2013

I was taught the 20 times table under a nun's watchful eye. Also, auxiliary verbs ("...maymightmustcanshould!"). And other crap I don't remember.

This article reminded me of a fallacy that seems to be applied most often to math education. We sometimes act like the the point of education is only to pour practical facts and methods into a child's brain, and the rest is a worthless waste of time.

But, for example, we ran windsprints in soccer practice. I don't ever remember anyone pouting and complaining about how we're never going to run windsprints in real life, or how we weren't even practicing running windsprints on the same field we played our games on. We pouted and complained for other reasons.

The point is not that I still remember the 20 times table (I don't), the point is, back then, putting in the effort to do so, and maybe it sharpened my mind and that carried on to other things (bonghit).

posted by fleacircus at 12:15 PM on June 27, 2013 [2 favorites]

This article reminded me of a fallacy that seems to be applied most often to math education. We sometimes act like the the point of education is only to pour practical facts and methods into a child's brain, and the rest is a worthless waste of time.

But, for example, we ran windsprints in soccer practice. I don't ever remember anyone pouting and complaining about how we're never going to run windsprints in real life, or how we weren't even practicing running windsprints on the same field we played our games on. We pouted and complained for other reasons.

The point is not that I still remember the 20 times table (I don't), the point is, back then, putting in the effort to do so, and maybe it sharpened my mind and that carried on to other things (bonghit).

posted by fleacircus at 12:15 PM on June 27, 2013 [2 favorites]

**DU**:

*Because I have a terrible memory for infrequently-used tricks, I prefer the more general trick to square any number.*

(a+b)

(a+b)

^{2}= a^{2}+ b^{2}+ 2ab**Emperor SnooKloze**:

*I remember reading some years ago about a trick to squaring numbers near 50.*

In a similar spirit, but more general, my favorite trick for mental multiplication is

(a+b)(a+c) = a(a+b+c) + bc.

Here, you get to choose "a" to make the calculation as simple as possible. Usually you want to set "a" to a nice round number close to one of the multiplicands.

For example, here's how I'd use this trick to calculate 63 times 78 (using a=60):

63•78 = 60•81 + 3•18 = 4860 + 54 = 4914.But you could also do it like this (with a=70):

63•78 = 70•71 - 7•8 = 4970 - 56 = 4914.Or even like this, setting a=41 to make the other factor a nice round 100:

63•78 = 41•100 + 22•37 = 4100 + 740 + 74 = 4914.For those who prefer words to algebra, here is a gloss in words:

To multiply two numbers, add whatever you want to one of them to make it simple, subtract the same amount from the other number, and multiply the two resulting numbers. Then you have to add or subtract a correction term, which is the product of the distances from one of the new numbers to the two original numbers. The correction is added if the new numbers are farther apart than the original numbers, or subtracted if they're closer together.

OK, it sounds kind of complicated that way. But with a little practice, it starts to make sense and becomes a very effective way to do mental math, not least because you can flex it to change the calculations to whatever you personally find easy (witness the three variations on the example above).

posted by aws17576 at 12:29 PM on June 27, 2013 [5 favorites]

*What is it about the number 12?*

It was a common base unit of counting for pastoralists and farmers. You could count a flock of sheep, or keep track of the numbers of rows hoed, by touching the thumb to each of the finger joints - this gives you twelve on one hand, and you keep track of the units of twelves by counting the finger joints on the other hand. You could keep track of up to 156 items with just two hands... with base ten, you can only count to ten with both hands.

posted by Slap*Happy at 12:32 PM on June 27, 2013 [1 favorite]

The 25 minutes/25 cents confusion makes a lot of sense because signatures.

I mean the fact that it is wrong is by-the-by, here-nor-there, beside-the-point. But then I suspect a lot of our 'certain' knowledge is the same - wrong, but answering some need. (Unless you're doing maths of course. I don't think there's a lot of here-nor-there in mathematics?

Having said that, a mathematician once told me maths as a language can be used to describe or express or conceive of anything and everything. I said 'yeah, right.')

*end/smartarse

posted by glasseyes at 12:43 PM on June 27, 2013 [1 favorite]

I mean the fact that it is wrong is by-the-by, here-nor-there, beside-the-point. But then I suspect a lot of our 'certain' knowledge is the same - wrong, but answering some need. (Unless you're doing maths of course. I don't think there's a lot of here-nor-there in mathematics?

Having said that, a mathematician once told me maths as a language can be used to describe or express or conceive of anything and everything. I said 'yeah, right.')

*end/smartarse

posted by glasseyes at 12:43 PM on June 27, 2013 [1 favorite]

Thank you, people, for teaching me the divided by 7 trick. I am now bitter I did not learn this in high school.

posted by jeather at 1:02 PM on June 27, 2013

posted by jeather at 1:02 PM on June 27, 2013

Metric or Imperial units, it doesn't matter. It's six of one, half a dozen of another.

posted by charlie don't surf at 1:50 PM on June 27, 2013

posted by charlie don't surf at 1:50 PM on June 27, 2013

Re: "12oz in a can of beer"

The new Budweiser bowtie can is 11.3oz for the same 12oz price. Plus, the geometry of those bizzles make them much harder to crush empties by stamping on them, which makes my job harder.

posted by Pirate-Bartender-Zombie-Monkey at 2:18 PM on June 27, 2013

The new Budweiser bowtie can is 11.3oz for the same 12oz price. Plus, the geometry of those bizzles make them much harder to crush empties by stamping on them, which makes my job harder.

posted by Pirate-Bartender-Zombie-Monkey at 2:18 PM on June 27, 2013

**Celsius1414**: /hits command-spaceHoly shit, spotlight does math! That's going to speed up my job a bunch. Thanks!

posted by Popular Ethics at 3:34 PM on June 27, 2013 [2 favorites]

When I was in grad school I decided that I was going to do everything in hexadecimal - I invented a relatively elegant way of saying hex integers, memorised the relevant addition and multiplication matricies, and was all set to really piss everyone off. Sadly I was defeated because for the life of me I can't do scientific notation in hex without resorting to mechanical assistance so in the end I had to fall back to insisting on using SI units for everything including time.

What I did learn from that little adventure is that choice of base is a compromise between compactness of expression and the size of the multiplication table you need to rote-memorise in order to work easily with integers.

posted by overyield at 3:34 PM on June 27, 2013 [1 favorite]

What I did learn from that little adventure is that choice of base is a compromise between compactness of expression and the size of the multiplication table you need to rote-memorise in order to work easily with integers.

posted by overyield at 3:34 PM on June 27, 2013 [1 favorite]

*What's 65 squared?*

As a developer, I know my powers of 2. So it's gonna be a little higher than 64

As a developer, I know my powers of 2. So it's gonna be a little higher than 64

^{2}= 4096.x

^{2}= (x−1)

^{2}+ x + (x−1)

65

^{2}= 64

^{2}+ 65 + 64

65

^{2}= 4096 + 129

65

^{2}= 4225

posted by Sys Rq at 3:58 PM on June 27, 2013 [2 favorites]

*As a developer, I know my powers of 2.*

So what is 2

^{53}?

Sure it's a big number, too big to write without scientific notation. In fact, it's such a big number that it's probably not necessary to know it exactly. If you could approximate it - at least getting close to the order of magnitude - then that's probably good enough. But how do you do that?

Easy! Since 2

^{10}= 1024 is approximately 1000 = 10

^{3}, you just do it like this:

2

^{53}= 2

^{3+10+10+10+10+10}=

^{3}2

^{10}2

^{10}2

^{10}2

^{10}2

^{10}

which is approximately

2

^{3}10

^{3}10

^{3}10

^{3}10

^{3}10

^{3}=

^{3}X 10

^{3+3+3+3+3}

^{15}

I've met developers who don't know this trick.

posted by twoleftfeet at 4:08 PM on June 27, 2013 [3 favorites]

overyield:

You know you can't just leave us hanging like that, right?

posted by ArgentCorvid at 4:52 PM on June 27, 2013 [1 favorite]

*I invented a relatively elegant way of saying hex integers*You know you can't just leave us hanging like that, right?

posted by ArgentCorvid at 4:52 PM on June 27, 2013 [1 favorite]

also, when writing a hex number in scientific notation, did you still use "x10^y"? or did you use a different symbol?

posted by ArgentCorvid at 5:00 PM on June 27, 2013

posted by ArgentCorvid at 5:00 PM on June 27, 2013

Computer folk have "hexadecimal literals", which are written like

0x1.23afp+14

which stands for

(1 + 2/16 + 3/(16*16) + 15/(16*16*16)) * 2^14.

It lets you specify the exact value of the binary representation. Without it, you can have a floating point number, and to send it somewhere you have to convert to decimal, then convert back. This saves you the conversion, and the accompanying problems with rounding that come with it.

posted by benito.strauss at 5:08 PM on June 27, 2013 [1 favorite]

0x1.23afp+14

which stands for

(1 + 2/16 + 3/(16*16) + 15/(16*16*16)) * 2^14.

It lets you specify the exact value of the binary representation. Without it, you can have a floating point number, and to send it somewhere you have to convert to decimal, then convert back. This saves you the conversion, and the accompanying problems with rounding that come with it.

posted by benito.strauss at 5:08 PM on June 27, 2013 [1 favorite]

The multiplication table is my weight room. That's the point.

posted by Renoroc at 5:13 PM on June 27, 2013

posted by Renoroc at 5:13 PM on June 27, 2013

What is wrong with you people?

Plus, this doesn't explain why you would need to learn the times

posted by delmoi at 5:20 PM on June 27, 2013

Why do you need to have the 12s memorized to figure that out? 20*3 = 60, and since we know 6*2 = 12 we know 6/(1/2) = 12, 6/0.5 = 12 and 60/5 = 12, so you need 5 dozen donutsIt involves the old english standard of the dozen. How many dozens of donuts do I need to feed twenty people if each person eats three? ?

Same thing 3*3 = 9, so 3*30 = 90, and 9*(4/3) = 12 so 9/(3/4) = 12 and 90/7.5 = 12. So you need 7 and a half dozen eggs.How many dozens of eggs do I need if I need to bake thirty cakes that need 3 eggs each

Plus, this doesn't explain why you would need to learn the times

*11*column.Which is exactly why you12 is better than 10 for a LOT of operations. Having roots of by 2, 3, 4 and 6 help show all sorts of the ways numbers interconnect to a young mind.

**don't**need to memorize it. They just need to multiply by 6 and double. Teaching kids to break numbers down to their problems to prime factors is what they need to learn. All these problems can be done using just the 10s time table. I actually think you only need to memorize the prime numbers in that table, since the rest you can just add. Might make things a little faster to memorize all of it I guess, but it isn't necessary.(x+y)Here's a quick trick. What's 65 squared? C'mon, what's 65 times 65?

^{2}= (x^{2}+ 2xy y^{2}) = (36*10*10 + 2*6*5*10 + 25) = 3600+600+25 = 4225Pff. Yeah if you like doing things the easy way.Do this. Take the digits before the last 5. In this case, that means 6. Multiply by the number one greater; in this case, that means 7. Now, 6 times 7 is 42. Then just tack on a 25 at the end.

That is, 65 times 65 is 4225.

2So what is 2^{53}?

^{64}/ 2^{11}. = (4 'gigabytes'^{2})/2^{11}, = (16 'exabytes')/2048 = 8 'petabytes'.posted by delmoi at 5:20 PM on June 27, 2013

It's east to tell whether numbers are divisible by 2 (is it even?), 3 (add the digits, divide by three), 4, 5, and 6 (work it out for yourself). I always used to get stuck on 7, 11, and 13 until I realised that 7•11•13 = 1001. So:

Take an arbitrary number, like 1234567.

Divide it into two parts at the third digit: 1234 567

Subtract the tail: 1234 - 567 = 667

(If you added 667 to 1234567 the sum would be evenly divisible by 1001)

Is 667 divisible by 7? No, because it's only 33 less than 700, and 33 isn't divisible by 7.

Is 667 divisible by 11? No because it's only 7 greater than 660.

Is 667 divisible by 13? No, because 50•13 is 650, which is 17 less than 667; and 17 isn't divisible by 13.

So 1234567 isn't divisible by 7, 11, or 13.

posted by Joe in Australia at 5:21 PM on June 27, 2013

Take an arbitrary number, like 1234567.

Divide it into two parts at the third digit: 1234 567

Subtract the tail: 1234 - 567 = 667

(If you added 667 to 1234567 the sum would be evenly divisible by 1001)

Is 667 divisible by 7? No, because it's only 33 less than 700, and 33 isn't divisible by 7.

Is 667 divisible by 11? No because it's only 7 greater than 660.

Is 667 divisible by 13? No, because 50•13 is 650, which is 17 less than 667; and 17 isn't divisible by 13.

So 1234567 isn't divisible by 7, 11, or 13.

posted by Joe in Australia at 5:21 PM on June 27, 2013

*Is there any point to the 12 times table? [more inside]*

posted by Cat Pie Hurts at 10:18 AM -

posted by Cat Pie Hurts at 10:18 AM -

**144 comments**YES!

posted by argonauta at 5:33 PM on June 27, 2013 [2 favorites]

*mippy, did you have trouble learning how to read an analogue clock face? I did, I think I was about 12 before I could. And I have to stop and collect myself in order to understand where is left and where is right.*

I am not mippy, but I commiserate nonetheless. Embarrassing but true: I still can't read an analog clock face quickly (thus why I'm happy to use cell phones and computers to tell time), I really

*cannot*tell my right from my left (and failed my first driving test due to it), I'm very ambidextrous (and since I don't know my right from my left, will often just give people whatever hand is closer when they ask for my dominant hand), and I also can't use buttons or zippers with normal dexterity (almost failed kindergarten due to that one). So, probably, dyspraxia.

Part ways with you on the math thing, though, as math is easy enough for me.

As for the 12 times tables, I find it amusing that so many of the folk justifications revolve around British money. I always assumed that it had to do with other systems like time, and wanting people to be comfortable with 12s since they're in use with those systems. It's been interesting to hear other, more solid justifications in this thread.

posted by librarylis at 9:50 PM on June 27, 2013 [1 favorite]

Getting back to the original article, it's too bad he stuck to just learning initial segments of the integers, i.e. from 1 to N for some N. Thinking about how I calculate, it seems like I've got 1 through 10, then for more accuracy I switch to 25, 15, 35, etc., staking a claim to the places half way between values I already know. I guess it's kind of like his friend who suggested learning the approximate reciprocals of 100.

posted by benito.strauss at 11:34 PM on June 27, 2013

posted by benito.strauss at 11:34 PM on June 27, 2013

Dozens are still somewhat relevant because 4*3 is a good way of packing things into a box. If you're counting stock in a store room, knowing the multiples of twelve still speeds things up.

Granted, on that basis knowing the mutliples of 16 is somewhat useful, but the mutiples of 11 is a waste of time, since nothing is ever packed in 11s. And not many people ever need to manually count stock in a store room anyway, but still...

posted by Segundus at 1:33 AM on June 28, 2013

Granted, on that basis knowing the mutliples of 16 is somewhat useful, but the mutiples of 11 is a waste of time, since nothing is ever packed in 11s. And not many people ever need to manually count stock in a store room anyway, but still...

posted by Segundus at 1:33 AM on June 28, 2013

I wrote a simple RPG that uses only one dice (a d12 because you could divide it so many ways) where instead of having rules and stats it had equations to calculate the rules based on certain attributes. I showed it to non-math RPG players (I managed to find a few) and they were flabbergasted that it could work but were surprised that it was somewhat intuitive to use once you had worked through your character sheet and calculated everything you needed to calculate.

They did think it was something like math homework though.

posted by koolkat at 2:07 AM on June 28, 2013

They did think it was something like math homework though.

posted by koolkat at 2:07 AM on June 28, 2013

*It's the same reason why we don't have decimalised time, or decimalised degrees in a circle.*

This comment has ten favourites. I hope it stays that way.

posted by obiwanwasabi at 2:13 AM on June 28, 2013

Also, Trachtenberg.

Unlike Lando, he's a system and a man.

posted by obiwanwasabi at 2:16 AM on June 28, 2013 [1 favorite]

Unlike Lando, he's a system and a man.

posted by obiwanwasabi at 2:16 AM on June 28, 2013 [1 favorite]

Happy Tau Day!

No. As someone who learned multiplication without tables, I can't wrap my head around it. I mean, it's good for getting people to pass tests by knowing values, but it doesn't teach you math. In fact, the best way to learn multiplication doesn't even use numbers (this is true of most good math). OK, so maybe not proveably the

Math is a conceptual process, not arithmetic. It's about relationships (among other things). The arithmetic approach is borderline useless, alienating, limiting, and generally a disservice to kids (and the adults they will become). If I had my druthers, math education would almost completely de-emphasize numbers until later in grade school. Let kids play with shapes, forms, and logic; human minds ADORE finding patterns (and doing so lights the brain up with all kinds of dopamine and whatnot that's just lovely).

posted by Eideteker at 6:44 AM on June 28, 2013 [4 favorites]

*"Is there any point to the 12 times table?"*No. As someone who learned multiplication without tables, I can't wrap my head around it. I mean, it's good for getting people to pass tests by knowing values, but it doesn't teach you math. In fact, the best way to learn multiplication doesn't even use numbers (this is true of most good math). OK, so maybe not proveably the

*best*way, but the way**I**learned it was by using dots on a grid. Multiplication is what you get when you draw a box this many dots by this many dots. Which is great, because it works in any (positional) number system. And it allows you to notice certain relationships (made it really easy for me to intuitively understand pyramidal numbers when my dad would take me with him to score his bowling practices).Math is a conceptual process, not arithmetic. It's about relationships (among other things). The arithmetic approach is borderline useless, alienating, limiting, and generally a disservice to kids (and the adults they will become). If I had my druthers, math education would almost completely de-emphasize numbers until later in grade school. Let kids play with shapes, forms, and logic; human minds ADORE finding patterns (and doing so lights the brain up with all kinds of dopamine and whatnot that's just lovely).

posted by Eideteker at 6:44 AM on June 28, 2013 [4 favorites]

Never underestimate the usefulness of basic skills such as numeracy and literacy.

posted by unSane at 7:19 AM on June 28, 2013

posted by unSane at 7:19 AM on June 28, 2013

The way arithmetic is taught might be alienating and limiting - but that doesn't mean it

posted by delmoi at 8:35 AM on June 28, 2013

*needs to be*. Think about the way music is taught in elementary school, compared to math. Is it any surprise everyone loves music and hates math? You can probably teach arithmetic in a way that's less*painful*and doesn't engender a life long dislike of mathematics.posted by delmoi at 8:35 AM on June 28, 2013

Now this is interesting news about arithmetic.

posted by Eideteker at 11:08 AM on June 28, 2013 [1 favorite]

posted by Eideteker at 11:08 AM on June 28, 2013 [1 favorite]

>did you still use "x10^y"? or did you use a different symbol?

Of course - how else would you express a power of sixteen in hex? The idea was to pretend that it was completely natural to work in that base, although obviously it's not for people with standard educational backgrounds.

If I could remember my system for saying the numbers I would have included it - it involved new words for the A-F numbers and for place value; this was all nearly 20

posted by overyield at 7:36 PM on June 29, 2013

Of course - how else would you express a power of sixteen in hex? The idea was to pretend that it was completely natural to work in that base, although obviously it's not for people with standard educational backgrounds.

If I could remember my system for saying the numbers I would have included it - it involved new words for the A-F numbers and for place value; this was all nearly 20

_{A}years ago so eh, dunno.posted by overyield at 7:36 PM on June 29, 2013

Has any one else noticed that, no matter what base you're using, if someone asks you "What base are you working in?", the correct answer is always "10"?

posted by benito.strauss at 1:09 AM on June 30, 2013 [2 favorites]

posted by benito.strauss at 1:09 AM on June 30, 2013 [2 favorites]

Not if it's base 1.

posted by Joe in Australia at 5:48 AM on June 30, 2013

posted by Joe in Australia at 5:48 AM on June 30, 2013

...there are 50 ways to leave your lover, but, if you have to divide anything, only twelve of them work.

posted by mule98J at 10:26 PM on June 30, 2013

posted by mule98J at 10:26 PM on June 30, 2013

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