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Automating Alphametic Puzzles: Discovering Solutions Through Signatures and Permutations

This document explores the fascinating world of alphametic puzzles, including the classic SEND + MORE = MONEY problem. It provides a step-by-step approach to solving these puzzles by treating letters as algebraic variables and calculating their signatures. The process automates the derivation of values for letters using permutations of digits. By generating all possible digit combinations, we can efficiently identify valid solutions. A Java implementation is also referenced, showcasing how automation can simplify solving complex puzzles.

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Automating Alphametic Puzzles: Discovering Solutions Through Signatures and Permutations

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  1. Alphametic Puzzles Pi Mu Epsilon Dessert Presentation April 10, 2006

  2. An Example Problem • How can the pattern SEND + MORE ------------- MONEY • Represent a correct sum, if every letter stands for a different decimal digit?

  3. Alphametics • Term coined by J.A.H. Hunter in 1955 • Also called cryptarithms by S. Vatriquant • Many alphametic puzzles may be solved by hand – but what if we want to automate the process?

  4. Approach to a solution... • Step 1: Collect all terms on left-hand-side of the equation • Example: FL + CA = FUN • Rewrite as: FL + CA – FUN = 0

  5. Approach to a solution... • Step 2: treat each letter as an algebraic variable with a value in [0,9] that contributes to the sum or difference by its value and position • FL = 10*F + 1*L • CA = 10*C + 1*A • FUN = 100*F + 10*U + 1*N

  6. Approach to a solution... • Step 3: solve the simultaneous equations • Yikes! • Is there an easier way…?

  7. Key Concept – A Letter’s Signature • Each letter in an alphametic puzzle has a signature that is obtained by substituting 1 for that letter and zero for all the others in the formula • Represents the contribution this letter makes to the overall answer • If we multiply all letter’s signatures by their values, a correct assignment of values will produce zero

  8. Example Signatures • F = 10 + 00 –100 = -90 • L = 01 + 00 – 000 = 1 • C = 00 + 10 – 000 = 10 • A = 00 + 01 – 000 = 1 • U = 00 + 00 – 010 = -10 • N = 00 + 00 – 001 = -1

  9. So….? • The problem now is to find all permutations a1… a10 of {0, 1, 2, …, 9} such that a1 s1 + a2 s2 + … + a10 s10 = 0 • So let’s generate all 10! Permutations and try each one to see if we have a solution!

  10. Java to the Rescue! • 10! would be a large number of possibilities to try by hand! • Let’s see a computer implementation of this idea: • http://www.muc.edu/~cindricbb/sp06/alphametics/TheApplet.html

  11. Reference • Knuth, Donald. The Art of Computer Programming, Vol. 4, Fascicle 2, Addison-Wesley, 2005, pp. 44-45.

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