Ken's Puzzle of the Week
Central Tendency

I pulled out some old matchboxes from my cupboard and counted the number of matches in each.  The median, mode, and mean of these counts were respectively 3, 4, and 5.  What is the smallest number of matchboxes I could have found?

Stated differently, find a set of non-negative integers with the given measures of central tendency (median, mode, and mean) and the fewest number of elements.

Extension: Find sets (if possible) for each of the six assignments of {mean, median, mode} to {3,4,5}.  Repeat for {2,3,4}.

Definitions:

• mean: (sum of all the elements in the set) divided by (the number of elements in the set). (arithmetic mean)
• median: the value/element which splits an ordered set into equal halves.  If the set has an odd number of elements, it is the middle element; if the set has an even number of elements it is the arithmetic mean of the two middle elements.
• mode: the element in the set which occurs most often.