Why Nd4 Most Likely Is the Solution - Searching Through 180,828 Mate in Two Problems - Part IV

It is time yet again to continue the search for the hardest mate in two problems. If you just joined or missed the first three parts you can find them here:

Searching Through 180,828 Mate in Two Problems - Part I

Mate in Two Clustering!? - Searching Through 180,828 Mate in Two Problems - Part II

What Is Difficult to Solve? - Searching Through 180,828 Mate in Two Problems - Part III

I’m nearing the end of the deep dive into the huge collection of mate in two problems. I have tried to construct a scoring model based on the variables that I have picked out: average nodes used and average time spent to find mate (for Stockfish 16). However, the top scores were often not those problems I considered very hard, and often they involved a Knight move. If I were to make a book out of it then the solution better be original and hard for humans to even consider.

I therefore decided I needed a final variable. The first move of the solution! I made a Python script that found the first move in the mate in two solutions, and then I ran the 17,000 problems that I had selected in the last newsletter.

With this data, I have now created a rarity score based on the coordinate and piece that moves. Here are the results visualized as a heatmap and chart:

As you can see the most common square is d4 and the most common piece is the Knight, so we can now make a new rule: when in doubt play Nd4!

You can maybe also apply these findings when you are solving mate in two problems in regards to what order you should start analyzing when you do not see clear candidate moves.

David Hjelle, Problem, 1967

Arnoldo Ellerman and Giorgio Guidelli, Good Companion Folder, 1920

This last problem is one with a higher rarity score! After finding all the variables I have now constructed a scoring model for hardness. I have experimented quite a bit and I think this one gives the best results. I have normalized the values from 0 to 1.

\( \text{Hardness score} = \text{Normalized time} \times \text{Normalized nodes} + \text{Normalized depth}^2 + \text{Rarity score} \)

If only I knew one of Kramnik’s mathematicians I would be able to explain why this formula works the best, but it does!

With the scoring system I now have a short list of 500 problems. I plan to now manually select the 100 best and publish them in a new book.

Merry Christmas to you and thank you for reading this series!🎄

/Martin