The puzzle goes as follows:

Each letter corresponds to a number, what is the result of the multiplication?

×  P

I'm not sure where to start with this puzzle. I know that each letter corresponds to a number, but I don't have any additional information about what those numbers might be. The only clue I have is the multiplication equation itself.

I've tried to approach the problem by breaking down the equation into smaller parts, but I haven't made much progress. Could the product of the second row of numbers (P * Q) end in a 9 because the last digit of the product is the same as the last digit of the result? Beyond that, I'm not sure what to do.

Here is the origin of the puzzle: https://cdn.discordapp.com/attachments/869145673197580298/1083855309942030346/PIL_-_sammanfattning_av_fragor_oscar_bakhouch.doc

  • $\begingroup$ This general type of puzzle is called a "Cryptarithm", and you can find a handful of resources on the web that explain how to solve cryptarithms. The general method is to list restrictions, such as how P*R can't be greater than 9 since it did not cause a carry, and then list all of the possible digits for each of those letters and eliminate them one by one. $\endgroup$ Commented Mar 11, 2023 at 3:00
  • $\begingroup$ This particular puzzle has three solutions. A few logical conclusions, such as P>1, will limit the number of cases to check, but otherwise this is just a process of elimination. $\endgroup$ Commented Mar 11, 2023 at 3:44
  • 3
    $\begingroup$ Turns out OP is having trouble because the puzzle is outright broken. (As also noticed by @DanielMathias above.) This means it's just a random combination or 8 letters, and as such, cannot constitute a copyrightable "work" by any stretch of the imagination. For this reason, I decided to waive our usual policy of demanding attribution before answering: in this case an attribution would only serve as undeserved publicity for bad puzzles. $\endgroup$
    – Bass
    Commented Mar 11, 2023 at 7:32

1 Answer 1


You start by gleaning information from the various aspects of the calculation.

  1. The last two digits of the first number are same, the last two digits of the product aren't. So there was a carry, which means P x Q > 10. (so P and Q both are at least 2.)

  2. P x R plus a carry (see above) equals Q, a single digit number. Combined with P > 1, R must be less than 5.

After that, it seems there's very little to deduce, so brute-force it is:

  1. If R were 4, then P would have to be 2, the carry 1, so Q would become 9. But 499x2=998, so this would make S equal to Q. So R is not 4.

  2. If R were 3, then P would have to be 2, the carry could only be 1, making Q into a 7. 377x2=754 is a valid solution.

  3. If R were 2, then P couldn't be 4: that would make Q at least 9, and the carry would be way too big. So P would be 3. Then 277x3=831 and 299x3=897 wouldn't work, but 288x3=864 is ok.

  4. Finally, it R were 1, then Q equals P + carry. The options are 144x3=432 (P equals S), 155x3 (carry too small), 155x4 (carry too big), 166x4=664 (P equals S), 177x4=708 (ok), 188x4 (carry too small), 188x5 (carry too big), and finally 199x5=995 (many repeats)

So there are three possible solutions, 377x2=754, 288x3=864, and 177x4=708, which explains why the solution isn't reachable by clever logical deductions alone.

Or in so many fewer words: you are having trouble because the puzzle is bad.


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