Technique for solving numerical cryptograms?

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Numerical cryptograms require assigning unique digits to letters to satisfy arithmetic equations, such as SEND + MORE = MONEY. A common approach involves setting up a system of equations based on the letter values, but this often results in an underdetermined system that lacks sufficient equations for a unique solution. Computational methods, particularly algorithms that can verify guessed solutions quickly, are recommended for solving these puzzles. The discussion highlights the challenges of deriving enough equations and suggests that modern computational techniques may provide the most efficient resolution. Overall, leveraging algorithms appears to be the best strategy for tackling numerical cryptograms effectively.
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I'm sure everyone here has seen one of these things some time or another. Where you're given something like...

SEND
+MORE
______
MONEY

...and then you have to figure out what each letter represents as a number in order to make the solution make sense, and no two letters can equal the same number, and no number in front can equal zero.


This particular gram that I posted I already know the answer for, but is there a specific way to solve these?

I was thinking about using a systems of equations somehow, maybe...

1000s + 100e + 10n + d + 1000m + 100o + 10r + e = 10000m + 1000o + 100n + 10e + y.

... however I'm not sure how I'd get enough other equations to substitute!

There's got to be an easy, mathematical way to do these. Anyone got some ideas?
 
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Mathematically you get an underdetermined system of equations which cannot be solved.
Computationally this is an example for a TM with oracle band, i.e. guessed numbers can be verified quickly.

Considering nowadays possibilities, an algorithm which checks solutions would be the method of choice.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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