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 Some thoughts by Federico Kereki)

There are five different combinations of the six numbers that can be chosen:

24578
24557
24558
25578
45578

Summing the total of all the numbers for each combination gives:

24578 = 9
24557 = 6
24558 = 7
25578 = 1
45578 = 3

Therefore the only possible combinations are 24558 or 25578, assuming that the numbers must sum to 0, 1, 4 or 7.

If the hundreds digit must be even, then for 24558 it must be either 2, 4, or 8; and for 25578 it must be either 2 or 8. This gives 36 possible arrangements of the digits for 24558, and 24 for 25578. Therefore the probability that the number is a perfect square is at max 1/60.

Posted by Iain on 2004-03-23 15:56:03