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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.

Source: Original.

Solution

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