Alphametic Puzzles

## Alphametic Puzzles

An alphametic puzzle (also sometimes known as a cryptarithm) is a type of puzzle where words are put together into an arithmetic formula such that digits can be substituted for the letters to make the formula true. Here are two examples, the first being one of the simplest and the second being probably the best known alphametic puzzle:

```    I       S E N D
+ B B     + M O R E
-----     ---------
I L L     M O N E Y
```

In the first, I = 1, B = 9, and L = 0.

There are two rules that must be followed for solutions:

• The leftmost letter can't be zero in any word.
• There must be a one-to-one mapping between letters and digits. In other words, if you choose the digit 6 for the letter M, then all of the M's in the puzzle must be 6 and no other letter can be a 6.

Around the time I finished graduate school (in computer science) I wrote a program to solve addition alphametic puzzles. It's now about ten years later, and my wife and I have just had a baby. I gave Kristin the SEND + MORE = MONEY puzzle to do while she was nursing our daughter. She had fun solving it, and I found a few more for her. I thought it would be interesting to augment the program (written in C++) I wrote before to search for puzzles given a set of words. It turned out to be pretty effective at finding new ones, so I'm making some of them available on the net for anyone who's interested. Incidentally, my wife and I have only solved a few of these by hand.

I've found lots of valid puzzles, but I've only listed the ones here that were at least a little more interesting than random related words. For example, I found 77 puzzles made out of state names. I've just included ones where several states added together equal the word "STATES".

All of the puzzles listed here have exactly one correct solution in base 10 (meaning the digits 0-9 are usable). Incidentally, I've made available a web-based puzzle solver in case you need some help.

The number in parenthesis after the puzzle is the estimated difficulty on a scale from 1 to 5. Note that this is determined via automation, so it doesn't correspond in all cases with the difficulty a human has in solving it. It's really a guess at how hard the puzzle is.

At the bottom of this page are some puzzles that are sentences or phrases.

```BILL + WILLIAM + MONICA = CLINTON (4)

IDAHO + NEVADA = STATES (4)
ALASKA + KANSAS + TEXAS = STATES (4)
HAWAII + MAINE + TEXAS = STATES (4)
IDAHO + MAINE + OHIO = STATES (4)
IDAHO + NEVADA + OHIO = STATES (5)
KANSAS + MAINE + NEVADA = STATES (5)
KANSAS + MAINE + OHIO = STATES (4)
HAWAII + IDAHO + IOWA + OHIO = STATES (4)
IDAHO + IOWA + OHIO + TEXAS = STATES (5)

COSTARICA + ERITREA + ESTONIA = COUNTRIES (5)
AUSTRIA + COSTARICA + CROATIA + IRAN = COUNTRIES (5)
AUSTRIA + COSTARICA + IRAN + RUSSIA = COUNTRIES (5)
COSTARICA + CROATIA + IRAN + RUSSIA = COUNTRIES (5)
COSTARICA + ERITREA + RUSSIA + TUNISIA = COUNTRIES (5)
AUSTRIA + COSTARICA + CROATIA + RUSSIA + TUNISIA = COUNTRIES (4)
COSTARICA + ERITREA + ESTONIA + IRAN + NAURU = COUNTRIES (5)
AUSTRIA + COSTARICA + ESTONIA + IRAN + NAURU + RUSSIA = COUNTRIES (5)

BEIGE + GREY = COLORS (2)
CORAL + ORANGE = COLORS (3)
GREEN + ORANGE = COLORS (3)
SIENNA + SILVER = COLORS (3)
AQUA + MAROON + SALMON = COLORS (4)
BEIGE + OLIVE + SILVER = COLORS (4)
BLACK + MAROON + SALMON = COLORS (5)
BLUE + CORAL + PURPLE = COLORS (4)
BLUE + OLIVE + SILVER = COLORS (3)
BLUE + PLUM + PURPLE = COLORS (3)
BROWN + CORAL + CYAN = COLORS (2)
BROWN + CORAL + SALMON = COLORS (4)
CORAL + CYAN + MAROON = COLORS (3)
CORAL + CYAN + ORANGE = COLORS (3)
CORAL + CYAN + SIENNA = COLORS (4)
CORAL + GREEN + SALMON = COLORS (5)
CORAL + MAROON + SALMON = COLORS (4)
CORAL + MAROON + TAN = COLORS (3)
CORAL + MAROON + TEAL = COLORS (4)
CORAL + OLIVE + SILVER = COLORS (3)
CORAL + OLIVE + VIOLET = COLORS (3)
CORAL + SIENNA + TAN = COLORS (4)
CYAN + GREEN + GREY = COLORS (2)
CYAN + GREEN + ORANGE = COLORS (3)
GOLD + OLIVE + SILVER = COLORS (4)
GREEN + ORANGE + SALMON = COLORS (4)
GREEN + ORANGE + TEAL = COLORS (4)
LIME + MAROON + SALMON = COLORS (4)
LIME + RED + SILVER = COLORS (2)
LIME + SALMON + SIENNA = COLORS (4)
LIME + SILVER + VIOLET = COLORS (4)
MAROON + NAVY + SALMON = COLORS (4)
MAROON + SALMON + SIENNA = COLORS (4)
MAROON + SALMON + TEAL = COLORS (4)
AQUA + CORAL + MAROON + SALMON = COLORS (4)
CORAL + CYAN + GREEN + GREY = COLORS (3)
CORAL + GOLD + GREEN + RED = COLORS (2)
CORAL + GREEN + MAROON + SALMON = COLORS (5)
CORAL + GREEN + ORANGE + SALMON = COLORS (5)
CORAL + GREEN + ORANGE + SIENNA = COLORS (5)
CORAL + GREEN + TAN + TEAL = COLORS (3)
CORAL + OLIVE + SILVER + TEAL = COLORS (4)
CYAN + GREEN + GREY + ORANGE = COLORS (4)
GOLD + GREEN + ORANGE + RED = COLORS (4)
OLIVE + SILVER + TEAL + VIOLET = COLORS (4)
CORAL + CYAN + MAROON + NAVY + SALMON = COLORS (5)
CORAL + GREEN + ORANGE + TAN + TEAL = COLORS (5)

CHESS + CHECK + KING = KNIGHT (3)
CHECK + MATE + KING = KNIGHT (3)
CHECK + KING + QUEEN = KNIGHT (4)

PACIFIC + PACIFIC + PACIFIC = ATLANTIC (3)

SATURN + URANUS = PLANETS (3)
SUN + URANUS + NEPTUNE = PLANETS (4)
MARS + SATURN + URANUS + NEPTUNE = PLANETS (5)
SATURN + URANUS + NEPTUNE + PLUTO = PLANETS (5)

GEMINI + VIRGO = CANCER (4)
GEMINI + LIBRA = CANCER (4)
TAURUS + PISCES = SCORPIO (4)
LEO + LIBRA + ARIES = PISCES (4)

DALI + MONET = RENOIR (2)
MATISSE + RENOIR = CASSATT (4)
RENOIR + SEURAT = GAUGUIN (3)
CASSATT + RENOIR = PICASSO (3)
MANET + SEURAT = RENOIR (4)
CASSATT + CEZANNE + MANET = MATISSE (4)
CASSATT + MATISSE + MIRO = PICASSO (4)
CASSATT + MATISSE + MONET = PICASSO (4)
DEGAS + INGRES + SARGENT + SEURAT = GAUGUIN (5)
CASSATT + MANET + MATISSE + MONET = PICASSO (4)
MANET + MATISSE + MIRO + MONET + RENOIR = CASSATT (5)
CASSATT + GAUGUIN = ARTISTS (4)
GAUGUIN + RENOIR = ARTISTS (3)
BOSCH + CASSATT + ESCHER = ARTISTS (4)
BRAQUE + RUBENS + SEURAT = ARTISTS (4)
CASSATT + COROT + INGRES = ARTISTS (4)
CASSATT + DEGAS + SEURAT = ARTISTS (4)
CASSATT + MANET + MONET = ARTISTS (3)
CASSATT + PICASSO + RENOIR = ARTISTS (4)
CASSATT + RUBENS + SEURAT = ARTISTS (5)
COROT + PICASSO + RENOIR = ARTISTS (4)
CEZANNE + MANET + MATISSE = ARTISTS (4)
DAVID + MATISSE + SEURAT = ARTISTS (4)
DEGAS + SARGENT + SEURAT = ARTISTS (4)
DURER + GAUGUIN + SEURAT = ARTISTS (4)
GOYA + INGRES + SARGENT = ARTISTS (3)
INGRES + MONET + RENOIR = ARTISTS (4)
INGRES + RODIN + SARGENT = ARTISTS (3)
MANET + RENOIR + SEURAT = ARTISTS (4)
MANET + RUBENS + SEURAT = ARTISTS (4)
CASSATT + COROT + DALI + DAVID = ARTISTS (3)
CASSATT + COROT + ESCHER + SEURAT = ARTISTS (4)
CASSATT + COROT + INGRES + SARGENT = ARTISTS (4)
CASSATT + COROT + MANET + MIRO = ARTISTS (3)
CASSATT + COROT + PICASSO + RENOIR = ARTISTS (4)
CASSATT + COROT + RENOIR + RODIN = ARTISTS (5)
CASSATT + INGRES + RENOIR + SARGENT = ARTISTS (4)
CASSATT + MANET + MIRO + RENOIR = ARTISTS (5)
CASSATT + MANET + MONET + RENOIR = ARTISTS (5)
CASSATT + MATISSE + MIRO + SEURAT = ARTISTS (4)
DAVID + DEGAS + INGRES + SARGENT = ARTISTS (4)
DAVID + DURER + MATISSE + SEURAT = ARTISTS (4)
DEGAS + DURER + INGRES + SARGENT = ARTISTS (4)
DEGAS + INGRES + MANET + MATISSE = ARTISTS (4)
DURER + MANET + MATISSE + SEURAT = ARTISTS (4)
GAUGUIN + INGRES + RENOIR + SEURAT = ARTISTS (5)
GOYA + INGRES + RENOIR + SARGENT = ARTISTS (4)
INGRES + MANET + MIRO + MONET = ARTISTS (3)
INGRES + MATISSE + MIRO + SARGENT = ARTISTS (4)
INGRES + MIRO + MONET + SARGENT = ARTISTS (4)
INGRES + RENOIR + SARGENT + SEURAT = ARTISTS (5)
MIRO + MONET + RENOIR + SARGENT = ARTISTS (5)
CASSATT + COROT + MANET + MIRO + RENOIR = ARTISTS (5)
CASSATT + COROT + MIRO + MONET + RENOIR = ARTISTS (5)
CASSATT + MANET + MIRO + MONET + RENOIR = ARTISTS (5)
COROT + MANET + MATISSE + MONET + RENOIR = ARTISTS (5)
COROT + MATISSE + MIRO + MONET + RENOIR = ARTISTS (4)
DURER + GAUGUIN + INGRES + SARGENT + SEURAT = ARTISTS (5)
INGRES + MANET + MATISSE + MONET + SARGENT = ARTISTS (4)
INGRES + MANET + MATISSE + SARGENT + SEURAT = ARTISTS (5)
INGRES + MANET + MIRO + MONET + SARGENT = ARTISTS (4)
INGRES + MATISSE + MIRO + MONET + RENOIR = ARTISTS (4)
MANET + MATISSE + MIRO + MONET + RENOIR = ARTISTS (4)
MANET + MATISSE + MONET + RENOIR + SARGENT = ARTISTS (5)
CASSATT + MANET + MATISSE + MIRO + MONET + RENOIR = ARTISTS (5)

SUMAC + ALDER + ELDER = TREES (4)
ALDER + MAPLE + ELDER = TREES (4)
ELM + ALDER + MAPLE + ELDER = TREES (4)
ALDER + MAPLE + ELDER + PALM = TREES (4)
ELM + ALDER + ASPEN + MAPLE + PALM = TREES (5)
ELM + ALDER + ASPEN + ELDER + PALM = TREES (5)
ALDER + ASPEN + MAPLE + ELDER + PALM = TREES (4)
ELM + ALDER + ASPEN + MAPLE + ELDER + PALM = TREES (4)

GOD + SODOM = BIBLE (2)
GOD + EVE + EVIL = BIBLE (2)
CAIN + SIN + MICAH + SARAH + CANAAN + ISAIAH + MANNA + NEHEMIAH + SACRAMENT = CHRISTIAN (4)

COMPLEX + LAPLACE = CALCULUS (2)
CIRCLE + ELLIPSE + EULER + SERIES + SUM = CALCULUS (2)
ABSCISSA + AREA + CIRCLE + EQUAL + EULER + SCALAR = CALCULUS (2)
AREA + CONE + LAPLACE + PARALLEL + POLAR + SCALAR + SLOPE = CALCULUS (3)
ARC + AREA + ELLIPSE + EULER + LAPLACE + SCALAR + SERIES + SUM = CALCULUS (3)

DNA + ALLELE + DIPLOID = SPECIES (4)
CLONE + GENE + MEIOSIS + MITOSIS = GENETIC (4)
DNA + ALLELE + CLADE + GENE = SPECIES (2)
```

I made a modification to my program to allow it to search text for sentences or phrases that are puzzles. I found several in the King James version of the Bible and some in the works of Shakespeare. Here are those and a few others that I found in other text:

```AND + HAD + NO = SONS (3)  (Numbers, 27:2)
AS + I + WAS + WITH = MOSES (2)  (Joshua, 1:5)
AND + UNTO + MOUNT = HERMON (2)  (Chronicles 1, 5:23)
THOUGH + HE + HATH = GAINED (2)  (Job, 27:8)
WHEN + THERE + WERE + NO = DEPTHS (3)  (Proverbs, 8:24)
WHOSE + TEETH + ARE + AS = SWORDS (4)  (Proverbs, 30:14)
THAT + THY + SHAME + MAY = APPEAR (3)  (Jeremiah, 13:26)
AND + ALL + HANDS + SHALL + BE = FEEBLE (3)  (Ezekiel, 21:7)
THEN + JESUS + WENT = THENCE (2)  (Matthew, 15:21)
AND + SAID + UNTO = SIMON (3)  (Luke, 7:44)
WHICH + WAS + WITH + THE = FATHER (2)  (John, 1:1)
STOOD + BEFORE + THE = THRONE (4)  (Revelation of Saint John the Devine, 7:9)

SINCE + JULIUS = CAESAR (4)  (Antony and Cleopatra)
NOT + THIS = YOUTH (2)  (CYMBELINE)
I + WILL + PAY + THE = THEFT (3)  (Hamlet)
I + THINK + IT + BE + THINE = INDEED (3)  (Hamlet)
THIS + IS + HIS = CLAIM (2)  (Henry V)
AND + NEVER + HEARD = BEFORE (3)  (Henry VI)
IF + THOU + DIDST + THIS + DEED + OF = DEATH (3)  (King John)
NO + MAN + BEARS + SORROW = BETTER (3)  (Julius Caesar)
WHAT + WAS + THY = CAUSE (2)  (King Lear)
ILL + MAKE + ALL = SPEED (3)  (Measure for Measure)
FOR + LACK + OF = TREAD (3)  (A Midsummer Night's Dream)
AND + THE + DEATH + OF + A + DEAR = FRIEND (2)  (A Midsummer Night's Dream)
HIS + HORSE + IS = SLAIN (2)  (King Richard III)
I + THINK + HE + CANNOT = CHOOSE (4)  (The Taming Of The Shrew)
HAD + I + A + GLASS + I = WOULD (3)  (The Taming Of The Shrew)
O + THOU + TOUCH + OF = HEARTS (2)  (The Life Of Timon Of Athens)
TIS + SAD + TITUS = CALLS (3)  (The Tragedy Of Titus Andronicus)

A + FAT = ASS (1)
AS + A = MOM (1)
I + DID = TOO (1)
OOOH + FOOD = FIGHT (1)

AFTER + TWO = YEARS (2)
AND + I + WONT + DO + IT = AGAIN (2)
BEER + AND = SODAS (2)
CALL + THEIR = OFFICE (2)
CHECK + THE = TIRES (2)
CUDDLY + PUPPY = MILITIA (2)
DO + YOU + FEEL = LUCKY (2)
EVER + SINCE = DARWIN (2)
FOR + LOST = GIRLS (2)
FULL + METAL = JACKET (2)
HAVE + A + GREAT = SUMMER (2)
HE + SEES + THE = LIGHT (2)
HES + THE = BEST (2)
HOW + MUCH = POWER (2)
I + KNEW + IT + WASNT + A = KEEPER (2)
I + LOVE + YOU = CEILE (2)
I + LOVE + YOU = EMILE (2)
ILL + STILL = LAUGH (2)
IT + ISNT + GOING + TO = HAPPEN (2)
IT + SUITS + MY = NEEDS (2)
MENTAL + HEALTH = MATTERS (2)
NO + PHONE + CALLS = PLEASE (2)
PLEASE + MAKE = OFFERS (2)
SADDAM + IS + A + MASS = MURDER (2)
SO + WHOS + THE = LOSER (2)
SO + WHY + WASTE + THE = EFFORT (2)
SO + YOU + WERE = RAPED (2)
SORRY + TO + BE + A + PARTY = POOPER (2)
SORRY + TO + BUST + YOUR = BUBBLE (2)
STEEL + BELTED = RADIALS (2)
STILL + HAS = TRUTH (2)
THAT + HIT + A = NERVE (2)
THATS + A + SIDE = EFFECT (2)
THE + WITCH + OF + THE = NORTH (2)
THIS + IS + TOO = FUNNY (2)
THROW + AWAY = PEOPLE (2)
TO + GO = OUT (2)
TWO + SHOWS + IN + ONE = NIGHT (2)
WE + ARE + DEAD = WORDS (2)
WHAT + A + GREAT = QUOTE (2)
WINTER + INTO = SPRING (2)
YOU + SILLY = GOOSE (2)
YOURE + FREE + TO = PURSUE (2)
ZERO + POINT = ENERGY (2)

AND + A + STRONG + OFFENSE + AS + A + GOOD = DEFENSE (3)
AND + HERES + A = STORY (3)
AND + PLANT + LITTLE = POSIES (3)
AND + THIS + ONE + DOESNT = EITHER (3)
BUT + HE + COULD = TEACH (3)
BUT + IT + STILL + IS = EARLY (3)
BUT + ITS + GOOD = MUSIC (3)
BUT + ITS + STILL = FUNNY (3)
COULD + ANYONE = REALLY (3)
DEATH + TO + ALL + THAT = APPOSE (3)
EASY + STREET + SIMI = VALLEY (3)
EVERY + THREE = MONTHS (3)
GET + NEAR + THE = GLASS (3)
HE + HAS + CLEAR = LUNGS (3)
HER + HOPE + BECAME + A = PRAYER (3)
HERES + THE + TRUTH = PEOPLE (3)
HOPE + THIS + HELPS + OTHER = PEOPLE (3)
I + AGREE + ITS = TOUGH (3)
I + DONT + THINK + ITS + THE = SEASON (3)
I + GUESS + THE + TRUTH = HURTS (3)
I + LIVE + VERY + VERY = WELL (3)
I + STILL + WANNA + STAY = ANALOG (3)
IF + SANTA + DID = EXIST (3)
INSIDE + THE + UNITED = STATES (3)
ISNT + THIS + FUN = STUFF (3)
IT + IS + NOT + A + DEAD = ISSUE (3)
IT + TAKES + A + LITTLE = EFFORT (3)
ITS + A + GREAT = STORE (3)
ITS + GOING + TO + BE + A = BLAST (3)
LETS + CUT + TO + THE = CHASE (3)
LETS + SEE + IF + IT = WORKS (3)
MALE + AND + FEMALE = SKULLS (3)
MEDIUM + SIZE = HOUSES (3)
NEVER + LEAVE + ME = ALONE (3)
NEW + IN + THESE = PARTS (3)
RIPPED + APART = INSIDE (3)
SLEEP + FOR = HOURS (3)
SOME + GOOD = IDEAS (3)
THAT + IS + WHY + IT + IS = FALSE (3)
THAT + ISNT + GOING + TO = CHANGE (3)
THAT + WAS + NOT + MONTY = PYTHON (3)
THE + CENTER + OF + THE = CIRCLE (3)
THE + CEO + COMES = FIRST (3)
THE + DEATH + OF + CANNON = FODDER (3)
THE + DEATH + VALLEY = DRIVER (3)
THE + DOG + GOT + HER + ON + THE + HAND = AGAIN (3)
THE + SECRET + IS = SIMPLE (3)
THE + UNDER = WORLD (3)
THERE + ARE = RISKS (3)
THERE + WAS + NO + OTHER = REASON (3)
THEY + DIDNT + RESIST = ARREST (3)
THIS + DOES + THE = TRICK (3)
UNDER + THE + UNITED = STATES (3)
USING + LOCAL = ACCESS (3)
WE + ARENT + BABIES = EITHER (3)
WE + GOT + THE + WRONG + ONE = AGAIN (3)
WHAT + A + WEEK + AT + NEWS + THIS + HAS = BEEN (3)
WHEN + I + REALLY + WANT + A = THRILL (3)
WHILE + LONDON = FLOODS (3)
WORLD + TRADE = CENTER (3)

BATMAN + GOTHAM = NIGHTS (4)
EASTERN + OREGON + STATE = COLLEGE (4)
ELVIS + PRESLEY = POSTERS (4)
GET + ALONG + LITTLE = DOGIES (4)
HIGHER + LEVEL = PEOPLE (4)
HOW + SWEET + COFFEE = TASTES (4)
I + GUESS + TASTES = CHANGE (4)
I + KNOW + I + DONT + WANT + TO + KNOW = THIS (4)
MAINTAIN + INTERNET = FIREWALL (4)
MIAMI + STILL + HAS + A = CHANCE (4)
THATS + ENOUGH = CHANGE (4)
THATS + THE + THEORY = ANYWAY (4)
THE + FOURTH + STATE + OF = MATTER (4)
THERE + ARE + OTHER + FORCES + OF = COURSE (4)
THESE + ARE + GREAT = DEALS (4)
THIS + IT + SEEMS + TO + ME + IS + THE + HEART + OF + THE = MATTER (4)
WHICH + IS + IN + ONE + CORNER + OF + HIS = OFFICE (4)
YEAH + THATS + OUR = STORY (4)
YOUVE + NEVER + BEEN + TO = EUROPE (4)
```

I don't have a particular interest in solving alphametic puzzles myself, but I did enjoy writing the program to solve and generate them. There are some other pages dealing with more exotic types of these puzzles. Mike Keith has created a page with interesting types of these puzzles, and Jorge Soares has a page with more puzzles and links.