Problem 20 · 2020 Math Kangaroo
Stretch
Counting & Probability
grid-countingspatial-reasoning
Ana plays with n × n boards by placing a token in each of the cells, with no common points with other cells containing tokens. In the picture we see how to place as many tokens as possible on 5 × 5 and 6 × 6 boards. In this way, how many tokens can Ana possibly put on a 2020 × 2020 board?
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Answer: D — 1010²
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Hint 1 of 2
'No common points' means tokens can’t even touch at a corner, so leave a gap between rows and columns.
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Hint 2 of 2
Place tokens on every other row and every other column; count how many fit.
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Approach: pack tokens on alternate rows and columns
- Non-touching tokens go on alternate rows and alternate columns.
- On an n×n board that fits ⌈n/2⌉ per direction.
- For 2020, ⌈2020/2⌉ = 1010 each way, giving 1010² tokens.
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