Water Bucket Conundrum This problem has crossed my path a number of times in various guises. Perhaps you have also seen a version of it. You are staying at a rural cabin, and the only method to get water is to draw it from a well. A 4-gallon bucket and a 9-gallon bucket are the only containers for carrying

Problems to Ponder: Neighborhood Sleuth

Place a number in each of the following empty boxes so that the sum of the numbers in any 3 consecutive boxes is 2013. What is the number that should go in the box with the question mark?

There are 4! = 24 ways to rank four objects. However, a friend told me that if ties are allowed, the number increases to 75.

I attempted to list all the possibilities by first listing the 24 orderings of four objects, then using brackets to group ties involving two players, then group ties involving three players, and finally the single case in which all four objects are tied. But something has gone wrong; my list includes just 69 possibilities, not 75.

What happened? Did I miss something, or was my friend mistaken?