Problem 27 · 2012 Math Kangaroo
Hard
Number Theory
perfect-squarecasework
Wanted are all three-digit numbers from 100 to 999 that have the following property: If you remove the first digit a square number remains and if you remove the last digit again a square number remains (e.g. 164 → (1)64 → 16(4)). How big is the sum of all numbers with this special property?
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Answer: D — 1993
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Hint 1 of 2
Both 'drop the first digit' and 'drop the last digit' must leave two-digit squares.
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Hint 2 of 2
The middle digit is shared — it is the units digit of one square and the tens digit of another.
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Approach: match shared middle digit
- The two-digit squares are 16, 25, 36, 49, 64, 81. The number a b c needs both 10a+b and 10b+c to be in this list.
- The shared digit b must be a units digit of one square and a tens digit of another, which works for b = 1, 4, 6, giving the numbers 816, 649, 164, 364.
- Their sum is 816 + 649 + 164 + 364 = 1993.
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