Why Is There Only One? A Puzzle That Runs Out

Take a hexagonal arrangement of cells built from rings around a centre, and fill the cells with the consecutive whole numbers from one upwards, using each exactly once. The requirement is that every straight line of cells sums to the same value, and because a hexagon has rows running in three directions rather than two, that means every row in each of those three orientations must reach the same total. The condition is therefore far more demanding than the square version, where only rows, columns and two diagonals are constrained. A hexagon of side two has nineteen cells and fifteen such rows, all of which must agree.

Exactly one non-trivial solution exists and its properties are worth stating:

• •It has nineteen cells, arranged in rings of one, six and twelve
• •It uses the numbers from one to nineteen, each once
• •Every one of its fifteen rows sums to thirty eight
• •It is unique apart from rotations and reflections of itself
• •A hexagon of side one, a single cell, works trivially
• •No hexagon of any larger size works at all

The impossibility is provable by a short argument that requires no searching, which is what makes the result satisfying. Counting the cells and the rows of a hexagon of any given size, and writing down the condition that the sum of all the numbers must be distributable equally across the rows in each direction, produces an equation relating the size to the required row total. Solving that equation for whole number answers yields only two possibilities, namely the trivial single cell and the nineteen cell case. Every other size produces a required row total that is not a whole number, which rules it out immediately. The search for larger examples is therefore not merely unsuccessful but unnecessary.

Comparing this with the far better known square arrangement shows why one is abundant and the other almost nonexistent. A square grid of a given size has rows, columns and two diagonals to satisfy, which is a substantial number of conditions and is satisfiable in enormous numbers of ways, with the count for a four by four grid running to thousands and for five by five to many millions. The hexagonal arrangement constrains three full sets of rows rather than two plus diagonals, which is proportionally far more demanding relative to the number of cells available. That difference in how tightly the conditions bind is the whole explanation, and it is a good illustration of how a small change in a rule can convert abundance into uniqueness.

The solution has been discovered independently several times, which is a recurring pattern with attractive small puzzles. Ernst von Haselberg published it in 1887. It was rediscovered repeatedly over the following decades by people unaware of the earlier work. Clifford Adams is widely reported to have spent nearly five decades from 1910 searching for it by hand with ceramic tiles, finding it in 1957, losing the paper he had written it on, and recovering it some years later, after which he learned it was already known. That story is frequently told as a caution about literature searches and is also a fair illustration of how much attention a genuinely unique object attracts.

Filling a hexagon with consecutive numbers so that every row in all three directions sums equally is a far stricter condition than the square version. Exactly one non-trivial arrangement exists, with nineteen cells, the numbers one to nineteen, and every row summing to thirty eight. Nothing larger is possible, provable by an equation that yields whole number answers only for those two cases.