Multiply a hundred sevens together. The number has eighty-five digits and nothing will display it. What is its last digit? One — and a child can get there in under a minute.
The answer key prints on a separate page.
Every other challenge sheet here is hard because something is missing. This one is hard because nothing is: the child has all the information and still cannot do the arithmetic, so the only way through is to notice that powers of 7 end in 7, 9, 3, 1 and then 7 again. Four values, forever. That shift — from computing to finding the pattern that makes computing unnecessary — is a large part of what number theory is.
Finding the cycle is the easy half. Landing on the right member of it is where the question bites, and it bites hardest when the division comes out even: 100 divided by 4 leaves nothing, so the hundredth power is the LAST of the four, not the first. A child who works out the remainder and counts from the start of the list gets 7 instead of 1. It is the fencepost error, arriving several grades later than usual and dressed up beyond recognition.