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there is a related problem (inspired by an interview questions from ITA software, a part of Google now) that asks to find the smallest positive integer that cannot be expressed using a digit K (in [2,9]) N times in an expression that involves the four basic arithmetic operations and the parentheses: +, -, *, /, (, and ).

to make the problem more interesting, the list of operations can be extended to include exponentiation and concatenation.

for example, using the four basic operations for N=4, and K in [2,9], we get:

7 cannot be computed in 4 2s; 11 cannot be computed in 4 3s; 10 cannot be computed in 4 4s; 8 cannot be computed in 4 5s; 9 cannot be computed in 4 6s; 4 cannot be computed in 4 7s; 5 cannot be computed in 4 8s; 4 cannot be computed in 4 9s;

The Claude C Compiler: What It Reveals About the Future of Software

"We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first ``local'' phase, a classical search algorithm is used to produce many desirable constructions. In the second ``global'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years."

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Thank you for your feedback. I am one of the authors of this paper.

I don't think the paper gives any hints on us dividing universities into classes and comparing. Would love to know how you reached that conclusion.

At the same time, yes, we could have divided universities into different classes: research vs liberal arts, this vs. that state, big vs. small size, etc. These are all trivial groupings but none of these would change the conclusions of this paper. For all practical purposes, we could easily have replaced the university names with labels like U1, U2, ... and still the conclusions would not change. What matters is how a weight-based composite index can be gamed and the paper does show that in multiple ways. Pls review the ILP formulations yourself and run them on the dataset of your choice.

Thank you for the comment. Pls refer to many references on the Web on the use of Fibonacci sequence as a dynamic programming example or illustration.

Thank you for the comment. Will check it out.