How to solve? (A bow to Polya)

“A problem well stated is a problem half solved” - Charles Kettering.

A couple of weeks back, June Huh, a Princeton Mathematics professor was one of the four mathematicians awarded the honor of 2022 Fields Medal. Quanta, NYT and several journals had beautiful profiles of the winners and their mathematical interests. Prof. Huh’s story felt atypical, it was a pleasant surprise to find that Prof. Huh was more interested in poetry and music through his high school and discovered a love for mathematics late during his university years. Prof. Huh mentioned a specific logic puzzle that triggered his interest in mathematics. Now here’s the puzzle, the task is to exchange the positions of the black and white knights (All the regular chess rules on knight movement holds)

Please take a crack at the puzzle before reading more. You don’t want to miss the magic, do you?

This puzzle’s promise emerges when one engages. Brute-force approach, i.e., multiple attempts, could solve the problem, but the puzzle offers more. You will notice that cells in the grid have different exit and entry options,

The entry & exit options at each cell indicates the flexibility that one has in moving the knights in or out. If you brute-force the problem, this flexibility is useful to keep in mind. But that’s not it! Let’s hear what Prof Huh had to say about his approach to research, “Recasting math problems by simplifying them and translating them in a way that makes a solution more obvious has been the key to many breakthroughs.” This is great advice in the context of a math or business problem, i.e., there’s value in investing time and resources in clarifying the shape of a problem.

Here’s an attempt at re-casting the knight problem, let’s first name the cells, A through J.

A knight has a limited set of moves from each cell, therefore let’s re-cast the above grid as a pathway for knight movement, for e.g., Knight can move from A to C or A to G, therefore A is connected to C or G, as represented below,

Now the solution is blindingly obvious, there’s no need to waste energy in remembering the complications of a 3-step jump. You just move the black horses out of the way to cells E & I. Then move one of the white knights to H and then switch positions of a black and white knight. Finally move the last black knight to the original positions of the white knights. Done and dusted!

Prof. Huh was seeking beauty in formulation of a problem. The same spirit can animate any business problem solving session and the attempt should be made to formulate a problem in a way that allows intuition to work.

Congratulations to the Fields Medal winners!

“A problem re-cast is the beginning of a solution” – June Huh’s Corollary!

{This note is dedicated to Penta D5. And you reader, if you know Prof. June Huh, please post a selfie with him here.}