Problem 29 · 2010 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
How many three-digit numbers have the property that their middle digit is the average of the two other digits?
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Answer: E — 45
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Hint 1 of 2
The middle digit is the average, so the outer two digits add to twice the middle - an even sum.
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Hint 2 of 2
Count hundreds- and units-digit pairs that have the same parity.
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Approach: count same-parity digit pairs
- b = (a+c)/2 needs a + c even, so a and c share parity; b is then fixed.
- Odd a (1,3,5,7,9) with odd c (1,3,5,7,9): 25; even a (2,4,6,8) with even c (0,2,4,6,8): 20.
- Total = 25 + 20 = 45.
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