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Problem 599

Geometry Difficulty 2.7 Find the answer CEMC Pascal · Canada · 2022

A 5 cm by 5 cm pegboard and a 10 cm by 10 cm pegboard each have
holes at the intersection of invisible horizontal and vertical lines
that occur in 1 cm intervals from each edge. Pegs are placed into the
holes on the two main diagonals of both pegboards. The 5 cm by 5 cm
pegboard is shown; it has 16 holes. The 8 shaded holes have pegs, and
the 8 unshaded holes do not.

How many empty holes does the 10 cm by 10 cm pegboard have?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

A 10 cm by 10 cm board has 9 rows of 9 holes, or $9 \times 9 = 81$ pegs in total.

Each hole on the 2 main diagonals has a peg in it.

There are 9 holes on each diagonal, with the centre hole on both
diagonals, since there is an odd number of holes in each row.

Therefore, the total number of holes on the two diagonals is $9 + 9 - 1 = 17$ .

This means that the number of empty holes is $81 - 17 = 64$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.