Problem 11 · AMC 8 Stretch
Core
Counting & Probability
Number Theory
pigeonholeparity
Write any 6 whole numbers into the 6 cells of a 2-row, 3-column grid. Show that you can pick a rectangle (2 of the 3 columns) whose 4 corner numbers add up to an even number.
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Answer: such a rectangle always exists
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Hint 1 of 4
Whether a sum is even or odd only depends on even/odd, not the actual numbers. Replace each number by E (even) or O (odd).
Still stuck? Show hint 2 →
Hint 2 of 4
Look at each column as a top/bottom pair. A column's SUM is either even or odd. Label each column 'even-sum' or 'odd-sum'.
Still stuck? Show hint 3 →
Hint 3 of 4
Now you have 3 columns sorted into just 2 labels ('even-sum' or 'odd-sum'). What does pigeonhole say?
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Approach: Pigeonhole on column-sum parity — 3 columns, 2 labels
- Only even/odd matters for whether a sum is even, so replace each number by E or O.
- Look at each of the 3 columns as a top/bottom pair and ask whether its two numbers add to an even or odd total. Label each column 'even-sum' or 'odd-sum' — just 2 labels (boxes) for 3 columns.
- Since \(3 > 2\), two columns share a label. If both are 'even-sum', the four corners total even + even = even; if both are 'odd-sum', they total odd + odd = even.
- Either way, that pair of columns forms a rectangle whose 4 corner numbers add up to an even number.
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