Solution to Problem 79

I received many submissions, but the only correct solution was sent in by Tim Kelley. Way to go, Tim!

Of the 999³ possible addition problems, there are

220³ - 3(55³) + 3(1000) - 1 = 10151874

which yield no carry. The probability is therefore 10151874/999³ ».01018239, or a bit over 1%.

What makes this problem tricky is that the numbers are restricted to lie between 1 and 999. The problem is much easier if the numbers are allowed to take on values between 0 and 999, so do that problem first.

First verify that there are 220 possible addition problems of three 1-digit numbers between 0 and 9 with no carry. (Just write out a list and count them!) Thus there are 220³ addition problems of three 3-digit numbers each between 0 and 999 with no carries, which explains the first term.

How many of these have one of the summands a zero? We count these next and subtract them out. This is the number of two 3-digit addition problems with no carries, of which there are 55 (look at your list). The zero summand can occur in one of three possible places, which explains the next term in the solution.

However, some of these have two of the summands a zero. These have been counted out too many times, so we count these and add them back in. There are 1000 such, and the two zero summands can occur in three possible ways, which explains the third term.

Finally, there is one addition problem we added back too many times, namely the one that has all three summands zero. Subtracting this back out explains the last term.

You are visitor number 2958 to this page.
Page last updated 13 January 2000.