When I was young, my Mom used to buy crossword puzzle magazines at the grocery store. She would do the crossword and the cryptograms, and I would do the other stuff: anagrams, logic puzzles, and of course number puzzles. My favorite was the kind with a long division problem where the digits have been replaced by letters. After you figure out the cypher and list the letters for 0 through 9 in order, they spell out a word or phrase, just to demonstrate that you got it right. Here’s an example:
KID COOK|LACKING LONGD OKALN OOADN LGABG LGCKK LDA
This says that LACKING divided by COOK gives KID with remainder LDA. We also know that
- COOK times K equals LONGD
- COOK times I equals OOADN
- COOK times D equals LGCKK
There are also some simple number facts that will come in handy:
- The rightmost column of a subtraction is simple. Say, if 8 minus Q is 2, then Q must be 6. If it’s not the rightmost column, Q might be 5, because the 8 might have been borrowed from.
- Pay attention to the ones digit in multiplications.
- If XYZ times Z ends in Z, then Z is 0, 1, 5 or 6.
- If XYZ times P, Q, and R all end in Z, then Z is 0 or 5.
- If XYZ times Q ends in Z, and Z isn’t 0 or 5, then either
- Q is 1, or
- Q is 6 and Z is even
OK, let’s start with the simple stuff. In the units place of the second subtraction,
N - N = B, so B is 0. In the hundreds place,
A - A = G. Since G isn’t also 0, G must be nine and there’s borrowing (D is bigger than L)
K * COOK ends in D, not in K, so K can’t be 0, 1, 5, or 6. On the other hand,
D * COOK ends in K, even though D is obviously not 1. So D is 6 and K is even. Since K*K ends in 6 and K isn’t 6, K is 4.
Let’s fill in what we have so far.
4I6 COO4|LAC4IN9 LON96 O4ALN OOA6N L9A09 L9C44 L6A
9 - 4 = A, A is 5. Since
A - O = O with no borrowing from the column to the left, O is 2.
L - 6 ends in 5, so L is 1. A borrowed-from 5 minus C is 1, so C is 3. Now we have:
4I6 3224|1534IN9 12N96 2451N 2256N 19509 19344 165
Almost done now.
I - 6 = 1, so I is 7, and a borrowed-from 3 minus N is 4, making N 8. The full division problem is
476 3224|1534789 12896 24518 22568 19509 19344 165
And the word is
Note that this took nothing beyond grade-school arithmetic.
Here’s another one to practice on. The first correct solution gets, um, bragging rights. Have fun, and show your work.
(Hint: Abgr gur havgf qvtvgf bs gur cebqhpgf.)
ELI MRO|MODELB RLLE RBFL RDAL OIFB ORLI ORM
By the way, there are only 10!, or about 3.6 million possibilities, so a computer program that brute-force tried all of them would run in well under a second. That would be cheating.