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.

(In reply to without a calculator by Charlie)

It turns out that a number with 49 as its last two digits can be a perfect square only if the hundreds digit is even.  This has probability 1/2 of happening.  However, if it is even, the probability that is a square is 8 times what it otherwise would be, leaving us with just the same probability of four times the most naive estimate, i.e., it is still 1/1,250,000,000.

Posted by Charlie on 2004-03-23 14:57:04