Hopscotch Possibilities

August123

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Two people want to play hopscotch. Draw all the hopscotch grids they could make with 4,5 squares. A specific number of squares can yield a total of 34 grids. Find this number.

Now I know the answer for this question, but I have got a problem: I don't know how to prove it other than drawing all the 34 grids!

I got the first part of the question easily because there were only 5 and 8 grids each. But to find the answer for the second part, it took me some time: 8 squares. The list of numbers of grids for the number of squares was this: 5,8,13,21,34. I can't find any kind of relationship between these numbers. Does anyone have an idea how to prove it?
 
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Two people want to play hopscotch. Draw all the hopscotch grids they could make with 4,5 squares. A specific number of squares can yield a total of 34 grids. Find this number.

Now I know the answer for this question, but I have got a problem: I don't know how to prove it other than drawing all the 34 grids!

I got the first part of the question easily because there were only 5 and 8 grids each. But to find the answer for the second part, it took me some time: 8 squares. The list of numbers of grids for the number of squares was this: 5,8,13,21,34. I can't find any kind of relationship between these numbers. Does anyone have an idea how to prove it?
What is the difference between a "square" and a "grid"? What is a "four,five square"? How did you get your answer for the first part? Thank you! ;)
 
The grid is made of squares, either in 1s or 2s. By drawing different possibilities, I got the answers 5 and 8:
1,1,1,1
1,2,1
2,1,1
1,1,2
2,2

1,2,2
2,2,1
2,1,2
1,1,1,1,1
2,1,1,1
1,2,1,1
1,1,2,1
1,1,1,2
 
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What is the difference between a "square" and a "grid"?

Google hopscotch grid, and you'll see images. These grids are composed of rows of squares (either 1 or 2 squares per row).


What is a "four,five square"?

I think that's a sloppy way of saying, "first make grids using 4 squares, then repeat using five squares." :roll:
 
The [total number] of grids [per] the number of squares [used is] this: 5,8,13,21,34.

That list starts with four squares used. Complete the beginning of your list, by considering 1, 2, or 3 squares used.


I can't find any kind of relationship between these numbers.

Once you have a list of grid counts for 1,2,3,4,5,6,7,8 squares used, look for a pattern. Big hint: consider pairs of adjacent numbers, in your list. :)
 
I got this: 1,2,3,5,8,13,21,34...
Grids with single squares only:
1,1,1,1,1,1
Grids with 1 two-square and single squares:
3,4,5,6,7,8
Grids with 2 two-squares and single squares:
1,3,6,10,15
Grids with 3 two-squares and single squares:
1,4,10
Grids with 4 two-squares:
1

These are the grids I got... For the grids with 3 two-squares and single squares, I thought that it would increase by 3,4,5,6 like the one before (2,3,4,5...) but it increased by 3,6,9... It's the same case for the grids with 4 two-squares (4,8,12...). How can I explain this?
 
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That's actually a very famous sequence called the Fibonnaci numbers. Consider each number in the sequence beginning with the third, and how you might write it as the sum of two other numbers. 3 = 2 + 1, 5 = 3 + 2, 8 = 5 + 3, etc. Now, if you're interested in a real challenge, perhaps try to explain why the "Hopscotch numbers" are the Fibonnaci numbers.
 
Thanks! I should've thought more about the "adjacent numbers" from Otis...
 
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