mezzacotta Puzzle Competition 2016

Solution: 2C. Some Faces

It should be clear that the first step is to identify the people whose faces and partial names are provided. Since we are sometimes given first names and sometimes given surnames, the relevant information to associated to each picture is likely to be the ungiven part of the name.

After identifying the more obvious people, we might notice that many of the names start with the same letter; in fact, many names share the same first two letters. On further investigation of these common beginnings, perhaps helped along by names like David Tennant and Sachin Tendulkar, we can see the common connection: Every name shares its first two letters with one of the numbers from one to eleven. Narrowing down the possible names in this way helps identify everyone; the complete list is:

Another feature that we might notice (that might help us spot the connection with numbers) is that there is one person whose name begins with "On", two with "Tw", and so on up to ten beginning with "Te". Then there is the outlier of Ellen DeGeneres, the sole eleven. Replacing each face with their corresponding number gives the following diagram.

After staring at this for a while, we might notice that the numbers are clustered in a suggestive way, forming groups that go up from some number to ten (or from ten to eleven, in one case). For instance, on the far left we have a vertical line that goes from 6 to 10, while on the far right we have a small box from 7 through 10. We can divide up the diagram in a fairly natural way into these separate clusters, one for each different starting number. If we connect the elements in each cluster with lines, connect-the-dots style, then we form the following picture:

This is a formula indicating one minus the sum of powers of a half. If we assume that the sum's first term is one half (corresponding to N = 1), then the formula evaluates to zero. We could intuit at this point that we want someone whose name would yield zero (i.e., someone whose name starts with "Ze"), but thinking about the formula a bit more laterally we can interpret it as representing half of something, then half of the remainder, and so forth — a conversion of the dichotomy paradox into mathematical form.

The answer to the puzzle is the inventor of that paradox, whose name would correspond to zero: ZENO OF ELEA. Another acceptable answer was simply ZENO.