# KenKen puzzles. Minimum number of “clues” to uniquely define nxn grid.

I recently discovered the "KenKen" puzzle and have been trying to figure out some of the mathematics behind it. This led me to the following question:

Given an N x N grid, what is the minimum number of filled-in spaces ("clues") needed to define a unique grid, where we must satisfy the requirement that every row AND every column contains exactly one of the integers 1 through N.

It's easy to gather that it depends not only on the number of clues, but on their values as well. For example, the following clues define a unique 3x3 grid

but a 3x3 grid with only 1's on the diagonal does NOT define a unique grid.

Does anyone have any insights into this (or other KenKen-related) problems? it is a very interesting topic indeed.

Thanks!

• So you are assuming you have no "cages" where a block of numbers have to add or multiply (or divide or subtract for a cage of two adjacent cells) to a given target? Cages are common in Kenken. – user2566092 Apr 18 '15 at 18:11
• Correct - This does not take into account cages. – Francis Apr 18 '15 at 18:30

What you are looking at is usually called latin squares in mathematics. These structures have the nice property of being the multiplication tables of quasigroups. A quasigroup is a set with a binary operation $(G,.)$ which has left and right division and cancellation. This means $\forall a,b \exists !c\;(a.c=b)$ and $\forall a,b \exists !d\; (d.a=b)$.