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 re: with a computer (or a table of squares) by Ady TZIDON)

You say:

<<you say: "This has probability 1/2 of happening" etc

Not true. The 5 tuple belonging to the original number is
either
8 5 4 7 2 or
8 5 4 5 2 or
8 5 5 7 2

All that out of "digits sum" consideration.
The prob. of the hundreds digit being even need to be evaluated for each of the cases along with the 5-tuples prob. as well. >>

Your possibilities are based on the number being a square.  The probability I gave of the 100's position being even was in general, not based on it's being a square. The probability was 1/2, as each of the 6 tiles was equally likely to have come from the 100's position (again, disregarding whether the number is a square or not), and half the digits are even. My calculation of the square probability, based on that alone, was then computed based on the necessity of the 100's position to be even in order to produce a square.

Posted by Charlie on 2004-03-23 20:58:57