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X-tracting a square root (Posted on 2009-03-14) Difficulty: 3 of 5
Solve this square root extraction problem, where each of the xs denote a decimal digit from 0 to 9, whether same or different. None of the numbers can contain a leading zero.

x x x x
√ x x x x x x 3 x
x x x
x x
x x x x
x x x x
x x x 3 x
x x x 3 x

No Solution Yet Submitted by K Sengupta    
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Solution solution Comment 1 of 1
The first digits are easy. At the start, the first two digits must be at least 10, and can't be higher than 16, as the highest square number smaller than these must be a single digit. This single digit has to be 9, so the first digit of the root is 3. In the next step, we have a 3-digit number with the largest value of (2*30+x)*x smaller than this number containing only 2 digits. The only value for x is 1, this gives the second digit of the root. The first two digits of the original 8-digit number must be 1 and 0.
All numbers below 3163 would only generate a 7-digit square, so we only need to check all numbers between 3163 and 3199. All multiples of 5 can be excluded, as they would generate a 2,7 or 0 as second-to-last digit in the square. A quick brute-force check of the other numbers shows there's only one root which has a square with a 3 as penultimate digit: 3194. Performing the root extraction to check shows this indeed does work.

  Posted by Robby Goetschalckx on 2009-03-14 14:46:18
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