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May 20

Solving 26 simultaneous equations

I have a puzzle that I want to solve where A thru Z each have a different value from 1 to 26 and I have a list of words for which the sum of each word's letters is given. Eg "BALL" = B + A + 2L = 8 might be solved with A=1 B=3 L=2 (values fixed). I know how to solve this in theory, but inverting a 26*26 matrix seems like hard work. Is there an easy way to solve/do this? Or even is there a website known that does this? Thanks. -- SGBailey (talk) 08:39, 20 May 2022 (UTC)[reply]

If A=1, B=2 and L=3, we get B + A + 2L = 9. We get to 8 by using A=1, B=3 and L=2. --Lambiam 10:38, 20 May 2022 (UTC)[reply]

Oh yes. Less haste, more accuracy and speed. -- SGBailey (talk) 12:29, 20 May 2022 (UTC)[reply]

See Gaussian elimination. Also, if you're lucky you may be able to find a subsystem of n equations involving only n unknowns (n < 26). Then you can solve this simpler system first and substitute its solution in the other equations. --Lambiam 10:30, 20 May 2022 (UTC)[reply]

Wolfram Alpha can invert matrices (see here), as can Microsoft Excel and LibreOffice Calc. AndrewWTaylor (talk) 13:01, 20 May 2022 (UTC)[reply]

Problem solved using excel's MINVERSE() function. Thanks. -- SGBailey (talk) 12:33, 23 May 2022 (UTC)[reply]