Alphametics

Mike Keith


 

This page is dedicated to that most elegant of puzzles (combining mathematical and word play) which as been one of my interests, on and off, for my entire adult life. If you've never seen an alphametic, I'll show you what the fuss is all about. If you have, I will try to regale you some of my own creations that have pushed the envelope of alphametic possibilities to new and bizarre heights.

Some History and Philosophy

An alphametic is a peculiar type of mathematical puzzle, in which a set of words is written down in the form of an ordinary "long-hand" addition sum, and it is required that the letters of the alphabet be replaced with decimal digits so that the result is a valid arithmetic sum. For an example one can do no better than the first modern alphametic, published by the great puzzlist H.E. Dudeney in the July 1924 issue of Strand Magazine:

 SEND
 MORE
-----
MONEY

whose (unique) solution is:

 9567
 1085
-----
10652

There are two fairly obvious (but worth stating) rules which every alphametic obeys:

1. The mapping of letters to numbers is one-to-one. That is, the same letter always stands for the same digit, and the same digit is always represented by the same letter.

2. The digit zero is not allowed to appear as the left-most digit in any of the addends or the sum.

Why are alphametics so cool? For one thing, there is their economy: with only a few words, a puzzle that can easily take half an hour to solve can be written down. The process of solving an alphametic is itself interesting, often illustrating the triumph of logic over trial and error. The puzzle above (SEND + MORE = MONEY) is especially elegant in this regard - it can be solved in a matter of seconds via a few observations.

The other obvious attraction is that alphametics are hard to construct. First of all, since we usually deal in base 10, only 10 different letters of the alphabet (at most) can be used. This, naturally, makes it hard to write phrases or sentences that read well. (There is a vague analogy here to the difficulty of writing a long palindrome that reads well.) Even if we write down a nice phrase or sentence representing a prospective alphametic, the odds that the alphametic will actually be solvable are pretty small. Finally, there is what I consider the most important feature an alphametic should have (but which imposes an additional harsh constraint on the constructor):

A truly elegant alphametic should have a unique solution.

This condition of uniqueness is often not required by alphametic constructors. They hack around this difficulty by presenting the problem with a "side condition", such as "make the sum a prime number" or some such statement. In my opinion this is very inelegant, and so on this page I only allow alphametics that have a unique solution.

The problem of alphametic construction can essentially be thought of as a very difficult form of constrained writing: the object is to write a phrase or sentence that (a) reads well, and (b) when considered as an alphametic (with the last word being the sum word), it is solvable (preferably with a unique solution). For the last ten years or so, I have primarily concentrated on this "meta-problem" - the construction of ever-more-elaborate alphametics that have a unique solution.

In the following sections, I'll present the creme de la creme of alphametic puzzledom - both traditional alphametics devised by others as well as some of my unusual creations. Every alphametic presented here has a unique solution, which you are hereby encouraged to find!
 

Some Traditional Alphametics

In this section I list some of the nicer "traditional" alphametics I've seen over the years. For conciseness, I just write them as sentences. It is understood that the first n-1 words are the addends and the nth word is the sum, that case is to be ignored, and that punctuation is not part of the alphametic (only the words). There are basically two types: phrases or lists, and complete sentences.

Phrases and Lists

Fifty states: America.       (Alister W. Macintyre)
Terrible number thirteen.       (Steven Kahan)
Earth, air, fire, water: nature.       (Herman Nijon)
Saturn, Uranus, Neptune, Pluto: planets.       (Peter J. Martin)
Georgia, Oregon, Vermont, Virginia.       (Sidney Kravitz)

The 'planets' one is very elegant, since it lists precisely the last four planets in the solar system.

Sentences

Winter breeze bred bitter freeze.       (M.R.W. Buckley)
Winter is windier, summer is sunnier.       (Brian Barwell)
No snow in view on roofs in Venice.       (A.G. Bradbury)
Martin Gardner retires.       (H. Everett Moore)
Nathan ate green peppers.       (Sidney Kravitz)
Amelia peeled a banana.       (Sidney Kravitz)
Who is this idiot?       (Sidney Kravitz)
Romans also more or less added letters.       (Steven Kahan)
Gee, I see a rare magic square.       (Steven Kahan)
Scientific American master creates frenetic interest in IMF metric (tens) state: fantastica!       (R.S. Johnson)

The last one here is a standout, because it reads well, is fairly long, and contains some long words. It also does a very nice job of not looking like it only contains 10 distinct letters. This alphametic inspired me to tackle the following interesting question: what's the longest word that can be worked into a uniquely-solvable alphametic? My best effort can be found in the New Literary Frontiers section.
 

The Doubly-True Genre

The doubly-true alphametic is an important sub-genre of alphametic puzzledom that made its first appearance in 1969. Since then, the creator of this first puzzle, Steven Kahan, has published literally hundreds more and has elevated the doubly-true alphametic to a high art form.

A doubly-true alphametic is one with the following remarkable property: the addends and the sum are "number words", and when read as words they also form a valid addition sum. Here is a simple example:

 THREE
 THREE
   TWO
   TWO
   ONE
------
ELEVEN

This, as a matter of fact, is the "smallest" doubly-true English alphametic with unique solution, where "smallest" means having the smallest sum word (11). In 1998 I conducted an exhaustive search of the 1,642,992,467 doubly-true alphametics with sum words TWO through ONEHUNDRED, and found that there are exactly 47764 with a unique solution. Here are some more examples:

SEVEN + SEVEN + SIX = TWENTY
EIGHT + EIGHT + TWO + ONE + ONE = TWENTY
ELEVEN + NINE + FIVE + FIVE = THIRTY
NINE + SEVEN + SEVEN + SEVEN = THIRTY
TEN + SEVEN + SEVEN + SEVEN + FOUR + FOUR + ONE = FORTY
FOURTEEN + TEN + TEN + SEVEN = FORTYONE
NINETEEN + THIRTEEN + THREE + TWO + TWO + ONE + ONE + ONE = FORTYTWO

Here are a few "long" examples. The last one is the longest (most number of addends), and also has the largest sum word (1000), of any uniquely-solvable doubly-true alphametic I'm aware of.

FOUR + THREE + THREE + THREE + THREE + ONE [24 times] = FORTY

FOURTEEN + THREE + TWO + ONE [22 times] = FORTYONE

NINETEEN + NINETEEN + TEN + TEN + TEN + TEN +
 NINE + NINE + NINE + NINE + NINE + ONE [877 times] = THOUSAND

New Literary Frontiers

In the "Traditional" section I present some examples of nice phrases or sentences that are uniquely-solvable alphametics. There's a vast, relatively unexplored, expanse beyond the humble sentence: longer sentences, poetry, special kinds of sentences (e.g., palindromes), and so on. It's not inconcievable that one could write an entire story in which each sentence is a uniquely-solvable alphametic. In this section I present some alphametics I've constructed that open up new territory in the literary realm.

First, a few poems. Here is a native nursery rhyme from the mythical island of Sevvoth that lies in the midst of the North Sea. In this poem, the words in the poem are the addends and the title of the poem is the sum word.

                Sevvoth

Ten herons rest near North Sea shore
    As tan terns soar to enter there.
As herons nest on stones at shore,
    Three stars are seen; tern snores are near!

Note that this poem has perfect meter. A different type of constraint is that imposed by the haiku, which consists of exactly three lines of 5, 7, and 5 syllables. Here is a haiku (again, with the title being the sum word) inspired by contemplating the flatness of glacial ice sheets:

                Flatiana

In Arctic terrain
An ancient, eerie ice tract
I enter a trance

Here is my best attempt so far to incorporate long words into an alphametic. This alphametic - a free-verse poem of sorts - contains a 17-letter word, which I believe is the current world record, along with a large number of other long words. Once again, the last word is the sum.

Realtor resales apparitions' sapiential stateliness,
    Entoparasite's reinterpretation piles settlor's interpenetrations;
Representational spoorers snap sanitationist's orientations.
    Representationist snootiness? Si. Snorers sin: sarsapirillas.
Retranslations, reinterpretation: "plenipotentiaries".

I once posed the following problem, which remained unsolved for several years: find a sentence that is a uniquely-solvable alphametic and also a palindrome. Finally, Mike Staniforth of the U.K discovered four examples, of which the two best are:

Emit a mile, Lima time.
Live Devo love devil.

and I subsequently found the following longer one, an extension of the classic Dennis and Edna sinned (in which "Dad-’n’-I" is interpreted as one word (DadnI) for the purpose of the alphametic):

Dennis, Nell, Edna, a Mr. Idi Amin, Dada, Dad-’n’-I, Maid Irma, and Ellen sinned.

One of my more complex creations is The Wiser Writer And The Inane Reader - the first medium-length poem written entirely in uniquely-solvable alphametics. Check it out.
 

"Found" Alphametics

In the late 1970's I constructed the uniquely-solvable alphametic

Double, double, toil, trouble.

which is nearly a quote from Act IV, Scene 1 of Macbeth, missing only the 'and' before 'trouble'. Many years later, this inspired the search for "found" alphametics - exact quotes from literature that just happen to be uniquely-solvable alphametics. Here are a few more I've found:

What was thy cause?       (King Lear, Act IV, Scene 4)
I think it be thine, indeed.       (Hamlet, Act V, Scene 1)

Chessametics

Lest you think that alphametic possibilities are limited to words, here is a rather unusual type of alphametic I invented about twenty years ago - the chess alphametic, or chessametic.

A good representative chessametic is the first one I published, in Vol. 8 No. 4 of the Journal of Recreational Mathematics (1975). Starting from the initial position in chess, consider the following legal chess game:

P-K4   P-K4
B-B4   P-R4
Q-B3   P-R4
QxP
......and checkmate: White says "I win!".

Amazingly, if we write all the moves of the game in a single column, with White's final exclamation at the bottom, we obtain a solvable alphametic, where the ten symbols to be replaced by digits are P,K,B,R,Q,I,W,N,-, and x:

P-K4
P-K4
B-B4
P-R4
Q-B3
P-R4
 QxP
----
IWIN

In a chessametic (or indeed, in any alphametic where some digits are already given), any digits already shown in the puzzle are simply to be left as is, and are still available to be substituted for one of the symbols.

This puzzle has two slight shortcomings from a chess point of view: (1) the two P-R4 moves are ambiguous (they could mean either KR4 followed by QR4 or the reverse), but this makes no difference in the final game position, and (2) in the last move (QxP), the fact that it is Black's KBP that is captured has to be inferred from the stated fact that this move is checkmate. These flaws are somewhat offset by the fact that this chessametic is of a very special kind, since it starts with the chess pieces in their initial position. It is, of course, much easier to construct a chess alphametic where the starting position as well as the moves are specified (as opposed to just the moves).

Here is another chessametic for your amusement. This one is much longer (12 moves!), also begins from the starting position, has no ambiguities in the chess notation, is a much nicer chess game, and as an alphametic has a unique solution. In short, it's perfect!

  P-K4    P-K4    
 N-KB3   N-QB3
  B-B4    N-B3
  P-Q4     PxP
   O-O    P-Q3
   NxP    B-K2
 N-QB3     O-O  
 P-KR3    R-K1
  R-K1    N-Q2
   BxP     KxB   
  N-K6     KxN
  Q-Q5    K-B3  
 Q-KB5

......and Black loses, saying "Boo hoo!"

The sum word, in this case, is BOOHOO .

Finally, here is an alphametic based on the game of checkers (a checkermetic). It uses algebraic notation to represent the moves, not the traditional notation in which the 32 squares are numbered. The starting position (w and b represent white and black pieces, not Kings) is:

        Black

 8    .   .   .   .
 7  .   .   .   .
 6    .   .   .   .
 5  .   .   .   .
 4    .   b   .   .
 3  .   .   b   .
 2    .   b   w   .
 1  .   w   .   w
    a b c d e f g h

        White

and the subsequent moves (White to move) are:

 g1-h2
 e3xg1
 c1xc5
CURSES

At this point Black CURSES because White has won. Again, the solution is unique.
 

Alphametic Squares

Here's another strange type of alphametic:

MANY
EGOS
ONCE
SEAR

This nxn array has an amazing property. If we attempt to solve it as an alphametic (with the bottom word being the sum), subject to the side condition of trying to maximize the numerical value of the sum, it has a unique solution. The same alphametic with the side condition "make the value of the sum minimum" also has a unique solution. If we rotate the array by 90 degrees, the resulting array also has a unique solution under each of the two side conditions! And so does the array rotated by 180 and 270 degrees. So, in all, there are eight interesting puzzles contained within this single array.

This is a 4x4 alphametic square. Do larger squares exist (e.g., 5x5)? Is it possible to make one that has words reading both horizontally and vertically (i.e., a traditional word square)? Is it possible to make one that has a single unique solution for all four puzzles (instead, as in this one, a unique solution for the conditions max and min sum)? These are all unsolved problems.
 

Order-n Alphametics

Here is a very bizarre kind of alphametic: the order-n cryptarithm. As an example, take the alphametic

TAR + STAR = TREE.

First, find its unique solution. Then, take each of the three numbers (the two addends and the sum) in the solution and replace each one (call it m) with f(m) = the last M digits of m (where M is the number of digits of m). For example, if one of the solutions numbers is m=234, you would square it to get 54756, then take the last three digits (756). Next, replace all the digits with letters, using the rule A=0, B=1, ... J=9. Consider what you have now as an alphametic, and solve it again! Then apply the function f(m) again, convert to letters, and solve again. And so on. In this remarkable puzzle, the first four alphametics in the chain (the original plus the next three derived ones) all have unique solutions (but the fifth one does not).

An alphametic of this type, in which n uniquely-solvable alphametics are cleverly concealed, is referred to as an order-n cryptarithm (under some function f(m), which must be specified). So TAR+STAR=TREE is an order-4 cryptarithm (whose four solutions you are hereby challenged to find!). Here are a few more, all of which use the same function f(m) as the one above:

COD + DAD = COOT (order 2)
MAY + RAG = RAJA (order 2)
DAB + AGAR = BAND (order 2)

So far I haven't devised any with order greater than 4, or with more than three words. But I'm sure they're out there waiting to be discovered...