Depth · one concept from first principles · step 2 of 4

Model two correspondences with separate rules

Game stones are placed following the pattern shown below. For the $30$ th figure, what is the difference between the number of white stones and the number of black stones?

The figures show stones arranged in growing squares. The 1st figure is a $3\times3$ array: the $8$ stones around the outer border are black and the single center stone is white. The 2nd figure is a $4\times4$ array: the outer border is black and the inner $2\times2$ stones are white. The 3rd figure is a $5\times5$ array: the outer border is black and the inner $3\times3$ stones are white. In this way, each step adds one more stone to a side.

Operations

Your answer

How to solve

Strategy Look for a Pattern — There are two separate correspondences hiding in one picture: how the white count grows and how the black count grows. I find a rule for each one separately by looking at the small figures, then use the two rules to compute each total for figure 30 and subtract.

1STEP 1

Find the rule for white stones

The white stones make an n by n block in figure n.

white(n) = n × n

2STEP 2

Find the rule for black stones

The black border is the (n+2) by (n+2) square less that white block, which comes to 4n + 4.

black(n) = (n+2)×(n+2) - n × n = 4n + 4

3STEP 3

Use both rules for the 30th figure

Apply each rule at n = 30. White is 30 x 30 = 900 stones. Black is 4 x 30 + 4 = 124 stones.

white(30) = 900, black(30) = 4 × 30 + 4 = 124

4STEP 4

Subtract to get the difference

There are more white stones than black, so the difference is white minus black: 900 - 124 = 776.

900 - 124 = 776

Answer

776

A check: white + black should be the whole 32x32 square. 900 + 124 = 1024 = 32 x 32, so the two counts fit together. The border is just a thin frame around a big white block, so white far exceeds black and a difference of 776 stones is sensible.

Takeaway

When one picture has two things changing, make a separate rule for each, then use both.

Find the rule for white stones
Find the rule for black stones
Use both rules for the 30th figure
Subtract to get the difference

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