Problems & Puzzles: Puzzles

Puzzle 179. Prime Alphametics

An Alphametic (also named cryptorithms) is an equation expressed in words that has some sense where each word is related to a number such that all the words and all numbers are congruent digit by digit.

Let's see an example in Spanish:

uno + uno + tres = cinco
563 + 563 + 9487= 10613

As a matter of fact the word sentence has sense because "uno+uno+tres" sums up to "cinco", and the numerical solution is also correct.

Moreover, the complete set of words and numbers used is congruent because:

for each two letters x & y in the conjunction word unounotrescinco and their corresponding digits m & n in the conjunction number 563563948710613, it happens that:

unounotrescinco
563563948710613

or x=y & m=n, or x<>y & m<>n.

By the way, the example written above, corresponds to the model more commonly studied:

 word1 + word2 +... = wordresult

This model has been so widely studied that you in the Web several solvers on line (*)

Necessarily the number of distinct letters used in any alphametic must not exceed 10 if the numbers are in base 10; and each word must be of the same size that the size of its associated number. Finally an alphametic is considered to be "good" if it has only one numeric solution.

Now on, we propose to study two new models and the magic-alphametic square.

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1) New model (studied by J. C. Meyrignac):

 word1 + word2 + ... wordk = wordk+1 + wordk+2 +...

Examples calculated by Jean-Charles:

Spanish: dos+dos+tres=siete (581+581+9231=10393)

English: two+two+two+two+seven=three+three+three+three+three

Italian: tre+cinque+cinque=tredici (184+923054+923054=1846292)

French:

deux+deux+deux+deux+deux+deux+deux+sept=trois+trois+trois+trois+trois+trois+trois

9842+9842+9842+9842+9842+9842+9842+5831=10675+10675+10675+10675+10675+10675+10675

For sure you must already have noticed that all the examples use words that are the names of prime numbers (on my request)

J.C. Meyrignac has sent his code written in C available on request to these pages.

Question 1. Can you find solutions to the Meyrignac model such that not only the words but the numbers associated to the words are all of them prime numbers

2) Model 2. SG-Model (suggested by CR)

 2 * word1 +/- 1 = word2

Example in Spanish: 2 * tres - 1 = cinco, 2 * 8209 - 1 = 16417

Notice that in this example the words are name of primes and the numbers are prime numbers.

Question 2. a) Can you find more solutions of this type? b) Can you find the largest solution in your language?

3) Magic Alphametic Square (suggested by CR)

 w1 w2 w3 w4 w5 w6 w7 w8 w9

Question 3. Can you find a set of 9 words (w1, w2, ..., w9) whose numeric solution is also a solution to a magic 3x3 square?

For this new object I have not still any specific solution. You are completely free to choose the nine homogeneous words (they can refer to numbers, to mathematicians, to planets, ...). Just make an elegant and ingenious election. Regarding the numbers associated to the 9 words, they aren't asked to be necessarily prime numbers.

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*Some links for this issue (given by J.C. Meyrignac):

Solution:

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