February 18, 2014 5:33 AM Subscribe

Visual Patterns. Here are the first few steps. What's the equation?

posted by Wolfdog (19 comments total) 26 users marked this as a favorite

posted by Wolfdog (19 comments total) 26 users marked this as a favorite

Any other computer programmers getting thrown off by assuming the first picture is Step 0?

posted by benito.strauss at 6:32 AM on February 18

posted by benito.strauss at 6:32 AM on February 18

If I saw the equations written out I'd never get past the first one, but seeing the drawings, the patterns are obvious. Which I assume is the point. Interesting, thanks for posting it.

posted by still_wears_a_hat at 6:45 AM on February 18

posted by still_wears_a_hat at 6:45 AM on February 18

The third diagram is 9 squares, 24 toothpicks.

posted by (Arsenio) Hall and (Warren) Oates at 6:47 AM on February 18 [1 favorite]

There isn't a single hard problem among these, as far as I can tell. Anyone who has practiced for certain Canadian government exams can do these in their sleep.

And the teacher's instructions state non-explicitly that Step #1 in the process is to turn the visualizations into numeric values.

posted by Yowser at 6:59 AM on February 18 [3 favorites]

And the teacher's instructions state non-explicitly that Step #1 in the process is to turn the visualizations into numeric values.

posted by Yowser at 6:59 AM on February 18 [3 favorites]

☃

☃

☃

Snowmen in step 43 = 1

What is the equation?

posted by 7segment at 7:21 AM on February 18 [3 favorites]

☃

☃

Snowmen in step 43 = 1

What is the equation?

posted by 7segment at 7:21 AM on February 18 [3 favorites]

By the way, most of these look to be polynomials at worst. There's a formal method (i.e. you could teach a computer how to do it, fairly easily if you're used to teaching computers how to do things) for finding the degree and then finding the polynomial. It's called "finite differences" and it's old school and pretty cool.

posted by benito.strauss at 7:50 AM on February 18 [5 favorites]

posted by benito.strauss at 7:50 AM on February 18 [5 favorites]

λx.☃

posted by idiopath at 7:53 AM on February 18 [4 favorites]

posted by idiopath at 7:53 AM on February 18 [4 favorites]

I love these because they temper my intellectual hubris.

posted by nerdler at 9:41 AM on February 18 [1 favorite]

posted by nerdler at 9:41 AM on February 18 [1 favorite]

Fun! I'm officially crap at maths but for what it's worth I found the first one harder than many of the ones that follow it. So don't be discouraged by #1.

posted by chrispy at 9:44 AM on February 18 [1 favorite]

posted by chrispy at 9:44 AM on February 18 [1 favorite]

These aren't advanced combinatorics exercises and won't tax the experts, but all in all I think it's a nice collection for just getting younger students thinking about all the different *kinds* of things you can count, and thinking in sequences: ok, here's a thing; what's a bigger / more complicated version of the same thing; what would one more step look like? And so on.

posted by Wolfdog at 10:24 AM on February 18 [1 favorite]

posted by Wolfdog at 10:24 AM on February 18 [1 favorite]

Mchelly: The great thing is that you have all the information to answer your own question. Certainly, sometimes puzzles like this would count 14 squares in step 3. But then they present you with the number of squares in step 43, so you can test the assumption.

If there are 9 squares in step 3, then the formula for number of squares is 3^2. So, if that's the right assumption, then there should be 43^2 squares in step 43. If the other assumption is correct, then there should be substantially more.

43*43=1849, which matches the number provided.

This gives us the answer to "What's the equation?" for number of squares: x^2

The equation for number of toothpicks is slightly harder, but it just takes a quick observation about shared edges of squares: x^2+2x

posted by 256 at 10:58 AM on February 18 [1 favorite]

If there are 9 squares in step 3, then the formula for number of squares is 3^2. So, if that's the right assumption, then there should be 43^2 squares in step 43. If the other assumption is correct, then there should be substantially more.

43*43=1849, which matches the number provided.

This gives us the answer to "What's the equation?" for number of squares: x^2

The equation for number of toothpicks is slightly harder, but it just takes a quick observation about shared edges of squares: x^2+2x

posted by 256 at 10:58 AM on February 18 [1 favorite]

Ok, so this is *exactly* how we introduce the concept of a variable in beginning algebra at my community college.

We give groups of students blocks and have them build the first few steps in the pattern, such as are shown in the link. Then we have them build the next two steps in the pattern, which are not shown. Next, we have them write down (in a complete sentence, or two) just*how* they would build, say, the 10th step in the pattern. This turns out to be the crucial step in the process, because different groups of students will just see it differently.

For example, in #107 from the link, you can imagine different ways of building the first one:*how* the builders are doing it...and writing that down. The upshot (as I see it) to having students build these things is that they have to be observant about a process (putting the blocks down) and clear in describing what is happening ("well, first she put down 4 blocks", etc.). The cool thing is that in an exercise like this, we often get four very different, but equivalent, methods which lead to very different looking expressions. Also, having these examples to look back on during the term can motivate some of the algebraic properties you look at later, such as the distributive property, difference of squares, etc.

I will leave it as an exercise for the reader to determine the four different "rules" for building #107 described above.

posted by klausman at 11:35 AM on February 18 [2 favorites]

We give groups of students blocks and have them build the first few steps in the pattern, such as are shown in the link. Then we have them build the next two steps in the pattern, which are not shown. Next, we have them write down (in a complete sentence, or two) just

For example, in #107 from the link, you can imagine different ways of building the first one:

- Put down 3 blocks on the bottom and top rows, then put 1 block on the left and the right. The next one would be built by putting 4 blocks on the bottom and top rows, then 2 blocks on the left and the right. The 10th would be built by putting 12 blocks on the top and bottom, then 10 blocks on the left and the right.
- Put down an "L" of 5 blocks, forming the bottom/left side of the picture, then add 3 blocks to fill in the top/right of the picture. The next one would be built by putting 7 blocks in an "L" shape to form the bottom/left side, then adding 5 blocks to complete the top/right side. The 10th one would be built by putting 23 blocks in the bottom/left side, and 21 blocks in the top/right side.
- Put a 3 by 3 square of blocks and remove the middle one. The next one would be built by building a 4 by 4 square of blocks and removing the middle 2 by 2 square. The 10th would be built by building a 12 by 12 square of blocks and removing the middle 10 by 10 square.

I will leave it as an exercise for the reader to determine the four different "rules" for building #107 described above.

posted by klausman at 11:35 AM on February 18 [2 favorites]

damn. I am so math-challenged that the question "what is the equation?" doesn't really even mean anything to me. As in I don't even know what they what me to think about.

posted by threeants at 5:20 PM on February 18

posted by threeants at 5:20 PM on February 18

threeants: they are asking about the pattern - first you can think of it as "how do you get from one step to the next", then you can think of it as "how do you construct step number _" where any number could fill in the blank (following the pattern, there is a way to jump to that step, just by knowing which step number it is)

posted by idiopath at 5:43 PM on February 18

posted by idiopath at 5:43 PM on February 18

No -because I'm a FORTRAN programmer.

posted by nightwood at 6:27 PM on February 18 [2 favorites]

I quickly learned that having an 11-yo sitting next to me made things a lot easier.

posted by sneebler at 6:58 PM on February 18 [1 favorite]

posted by sneebler at 6:58 PM on February 18 [1 favorite]

The equation for number of toothpicks is slightly harder, but it just takes a quick observation about shared edges of squares: x^2+2x

Hmm, I got s=x^2 and t=2(x^2+x), where the final equation is t=2(s+sqrt(s)). The math checked out too using the number 43.

This is where s is the #squares and t is the #toothpicks.

posted by womandad at 11:17 AM on February 20

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Is there an answer key anywhere so you can learn from this, or are you on your own

oh god please don't tell me this is aimed at fourth graders

posted by Mchelly at 6:20 AM on February 18 [2 favorites]