You have created a 19 digit number with your 20 digit tiles as follows:

7_340_46_2010_51_49

Unfortunately someone knocked out 5 of the number tiles and placed them with the remaining number tile. The 6 tiles that are out are 6 3 2 8 9 3.

Without using any calculators, programs, or similar devices, what is the easiest way to figure out what the probability that my original 19-digit number will be a perfect square and what is this probability?

Note: The pesky "someone" also renumbered the number tiles so they were wrong and you couldn't tell what the right numbers are. This has been fixed to make the problem easier.

If the number is a square, the sum of its digits should be 0, 1, 4, or 7.

The 14 digits sum to 1.

If we picked 8, 5, 4, 7 and 2, the sum would be 0, so the number could be a square.

If we picked 8, 5, 4, 7 and 5, the sum would be 3, so it couldn't be a square.

And so on...

Note, however, that this is a feasibility check; even if the sum is one of the four values, it can easily be not a square.

Posted by Federico Kereki on 2004-03-23 15:12:39