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. Years later when my wife and I had a baby, I gave her 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 add to 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, 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.

If you'd like to find some of your own puzzles, try my alphametic generator.

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.

## Puzzles that are sentences or phrases

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)

AS + A = MOM (1)
I + DID = TOO (1)
NO + NO + TOO = LATE (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)
DAYS + TOO = SHORT (2)
DO + YOU + FEEL = LUCKY (2)
EVER + SINCE = DARWIN (2)
FOR + LOST = GIRLS (2)
FULL + METAL = JACKET (2)
GONE + ON + THE = FRITZ (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 + NOT + THAT = AGAIN (2)
NO + PHONE + CALLS = PLEASE (2)
NOW + WE + KNOW + THE = TRUTH (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)
WHERE + IS + THE = TRUTH (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)
BUS + ERROR + CORE = DUMPED (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)
HOCUS + POCUS = PRESTO (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 + IS + STILL + BEING = TESTED (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)
PROOF + READER = DELUXE (3)
RIPPED + APART = INSIDE (3)
SEEST + THOU + THESE = SCALES (3)
SLEEP + FOR = HOURS (3)
SOME + GOOD = IDEAS (3)
STILL + WITHIN = LIMITS (3)
TASTE + ALL + THESE = DISHES (3)
THAT + IS + WHY + IT + IS = FALSE (3)
THAT + ISNT + GOING + TO = CHANGE (3)
THAT + WAS + NOT + MONTY = PYTHON (3)
THE + CATCH + AS + CATCH + CAN = MINDED (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 + NEVER + BEEN + A = SECRET (4)
THATS + THE + THEORY = ANYWAY (4)
THE + FOURTH + STATE + OF = MATTER (4)
THERE + ARE + OTHER + FORCES + OF = COURSE (4)
THESE + ARE + GREAT = DEALS (4)
THESE + TWENTY = NOBLES (4)
THIS + IT + SEEMS + TO + ME + IS + THE + HEART + OF + THE = MATTER (4)
UNITED + STATES + AND = CANADA (4)
WHICH + IS + IN + ONE + CORNER + OF + HIS = OFFICE (4)
YEAH + THATS + OUR = STORY (4)
YOUVE + NEVER + BEEN + TO = EUROPE (4)

## Puzzles involving related words

BILL + WILLIAM + MONICA = CLINTON (4)

Some puzzles with words from the Harry Potter books.

DEAN + DRACO + WAND = WIZARD (2)
RUBEUS + SIRIUS = AZKABAN (3)
DEAN + FRED + GEORGE = POTTER (3)
DEAN + MERLIN + WAND = WIZARD (3)
GEORGE + GINNY + RON = POTTER (4)
HARRY + CHARM + GINNY + RON = MAGIC (4)

States, countries, oceans, and planets.

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)

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)

Colors.

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)

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)
JESUS + BIBLE = CHURCH (4)
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)

## Long words

These puzzles are made up of words that are all at least 12 letters long.

4 Summands:

ARISTOCRATIC + PRESCRIPTION + PROSOPOPOEIA + PROTECTORATE = TRANSOCEANIC (3)

5 Summands:

NARRAGANSETT + NONOGENARIAN + OSTEOPOROSIS + PROPORTIONATE + TRANSPOSITION = RETROGRESSION  (5)
INTERNESCINE + OSTEOPOROSIS + PERSPIRATION + POSTPROCESSOR + PROSCRIPTION = TRANSPIRATION  (5)
ANTIPERSPIRANT + INTERSTITIAL + PRESENTATION + PROPORTIONATE + TRANSLITERATE = TRANSPORTATION  (4)

7 Summands:

ARISTOCRATIC + POSTPOSITION + PROCRASTINATE + PROPRIOCEPTION + PROSCRIPTION
+ PROTECTORATE + TRANSOCEANIC = INTERPRETATION  (4)
INAPPROPRIATE + OSTEOPOROSIS + PROCRASTINATE + PROSOPOPOEIA + RECONNAISSANCE
+ SPECTROSCOPE + TRANSCRIPTION = PROPRIOCEPTION  (5)

8 Summands:

APPERCEPTION + ARISTOCRATIC + INTERPRETATION + PRESENTATION + PROPRIOCEPTION
+ PROSCRIPTION + TRANSPIRATION + TRANSPOSITION = CONCESSIONAIRE  (5)
PERSPIRATION + POSTPROCESSOR + PRACTITIONER + PROSCRIPTION + PROSOPOPOEIA
+ RECONNAISSANCE + TRANSOCEANIC + TRANSPOSITION = CONCESSIONAIRE  (5)
POSTPOSITION + PRESENTATION + PROTESTATION + RECONNAISSANCE + SPECTROSCOPE
+ SPECTROSCOPIC + TRANSCRIPTION + TRANSPOSITION = INTERPRETATION  (5)
APPERCEPTION + ARISTOCRATIC + INCONSISTENT + OSTEOPOROSIS + PRESCRIPTION
+ PROTECTORATE + PROTESTATION + STATISTICIAN = SPECTROSCOPIC  (5)
INCONSISTENT + OSTEOPOROSIS + PERSPIRATION + POSTPOSITION + PRESENTATION
+ PROPORTIONATE + SPECTROSCOPIC + TRANSPIRATION = TRANSPORTATION  (5)
APPERCEPTION + INTERCEPTION + PRACTITIONER + PROSCRIPTION + PROTECTORATE
+ SPECTROSCOPE + SPECTROSCOPIC + TRANSOCEANIC = TRANSPOSITION  (5)

18 Summands:

APPERCEPTION + ARISTOCRATIC + CONCESSIONAIRE + CONSCRIPTION + INAPPROPRIATE
+ INCAPACITATE + INCONSISTENT + INTERCEPTION + OSTEOPOROSIS + PERSPIRATION
+ PRESCRIPTION + PROSCRIPTION + PROSOPOPOEIA + PROTECTORATE + PROTESTATION
+ STATISTICIAN + TRANSOCEANIC + TRANSPIRATION = ANTIPERSPIRANT  (5)

## Doubly True Alphametics

There is a class of alphametic puzzles involving spelled numbers which are doubly true. There is a vast number of these puzzles if you allow duplication of numbers (45842 with sums less than or equal to 100), but few where all of the numbers are unique. Here are all twelve of these in english with sums up to 100 (with exactly one solution):

ONE + NINE + TWENTY + FIFTY = EIGHTY (4)
TWO + FOURTEEN + FIFTEEN + TWENTY = FIFTYONE (4)
THREE + SEVEN + TEN + TWENTY + THIRTY = SEVENTY (4)
THREE + NINE + TEN + FOURTEEN + FIFTEEN = FIFTYONE (4)
FIVE + SIX + SIXTEEN + NINETEEN + TWENTY = SIXTYSIX (3)
SIX + NINE + ELEVEN + SIXTEEN + NINETEEN = SIXTYONE (3)
EIGHT + TEN + SIXTEEN + SEVENTEEN + EIGHTEEN = SIXTYNINE (5)
ONE + FIVE + TEN + ELEVEN + NINETEEN + FORTYFIVE = NINETYONE (3)
ONE + SEVEN + NINE + FIFTEEN + SIXTEEN + SEVENTEEN = SIXTYFIVE (5)
FIVE + SEVEN + ELEVEN + TWELVE + FIFTEEN + TWENTY = SEVENTY (5)
SIX + SEVEN + NINE + TWELVE + SIXTEEN + TWENTY = SEVENTY (5)
ONE + TWO + FIVE + NINE + ELEVEN + TWELVE + FIFTY = NINETY (5)

I've included some with repeated numbers. I've used a shorthand so these don't get too long by writing, for example, 5(two) instead of two + two + two + two + two.

SIX + SEVEN + SEVEN = TWENTY (2)
TEN + TEN + FORTY = SIXTY (3)
ONE + NINE + TWENTY + FIFTY = EIGHTY (4)
TWO + FOURTEEN + FIFTEEN + TWENTY = FIFTYONE (4)
THREE + THREE + THREE + ELEVEN = TWENTY (4)
FIVE + FIVE + NINE + ELEVEN = THIRTY (4)
FIVE + FIVE + THIRTY + FORTY = EIGHTY (5)
FIVE + FIVE + THIRTY + FIFTY = NINETY (3)
SIX + EIGHT + EIGHT + EIGHT = THIRTY (3)
SEVEN + SEVEN + SEVEN + NINE = THIRTY (4)
SEVEN + TEN + TEN + FOURTEEN = FORTYONE (4)
5(TWO) + 3(TWENTY) = SEVENTY (3)
5(FOUR) + TEN + TWENTY + FORTY = NINETY (5)
6(SEVEN) + EIGHT + THIRTY = EIGHTY (5)
6(TWO) + EIGHT + TEN + FIFTY = EIGHTY (2)
8(THREE) + FOUR + FOURTEEN = FORTYTWO (5)
4(EIGHT) + 4(NINE) + 2(ELEVEN) = NINETY (3)
8(SEVEN) + 3(EIGHT) = EIGHTY (3)
12(FIVE) + 2(SIX) + EIGHT = EIGHTY (3)
FIVE + 13(SIX) + SEVEN = NINETY (3)
85(ONE) + TWO + THREE = NINETY (5)

Below are some in foreign languages. I've ignored accent marks.

There are no doubly true alphametics in French with sums less than or equal to 100 where no numbers are repeated. Here are some where there are repeated numbers. As with all of the puzzles on this page there is exactly one solution to each of these.

UN + UN + NEUF = ONZE (1)
CINQ + CINQ + VINGT = TRENTE (3)
UN + UN + QUATRE + DOUZE + DOUZE = TRENTE (5)
UN + DEUX + DEUX + DEUX + DEUX = NEUF (2)
UN + QUATRE + CINQ + CINQ + QUINZE = TRENTE (4)
3(TROIS) + CINQ + SEIZE = TRENTE (4)
3(QUATRE) + NEUF + NEUF = TRENTE (4)
5(QUATRE) + CINQ + CINQ = TRENTE (4)
15(DEUX) = TRENTE (2)
2(UN) + 6(DEUX) + 6(SIX) + 5(DIX) = CENT (3)
9(UN) + 5(DEUX) + 7(DIX) + ONZE = CENT (3)
88(UN) + 6(DEUX) = CENT (3)

In Italian there are eleven doubly true (single solution) alphametics with sums up to 100 where there are no repeated words.

VENTITRE + TRENTA + TRENTOTTO = NOVANTUNO (3)
UNO + QUATTRO + SETTE + VENTOTTO = QUARANTA (4)
DUE + SETTE + DIECI + UNDICI = TRENTA (5)
TRE + SEI + DIECI + UNDICI = TRENTA (4)
UNO + NOVE + VENTIDUE + VENTOTTO + TRENTUNO = NOVANTUNO (4)
OTTO + NOVE + TREDICI + VENTI + TRENTA = OTTANTA (4)
DUE + SETTE + OTTO + NOVE + TRENTADUE + TRENTATRE = NOVANTUNO (3)
SEI + SETTE + DODICI + DICIOTTO + VENTISEI + VENTINOVE = NOVANTOTTO (2)
UNO + OTTO + NOVE + DIECI + UNDICI + VENTI + VENTUNO = OTTANTA (4)
DUE + TRE + SETTE + OTTO + NOVE + VENTUNO + TRENTA = OTTANTA (4)
UNO + DUE + TRE + OTTO + UNDICI + DODICI + TREDICI + TRENTA = OTTANTA (4)

Here are some Italian ones with repeated words:

TRE + CINQUE + CINQUE = TREDICI (3)
CINQUE + CINQUE + SEI = SEDICI (3)
UNO + CINQUE + CINQUE + CINQUE = SEDICI (3)
DUE + DUE + CINQUE + SETTE = SEDICI (4)
DUE + DUE + CINQUE + DODICI = VENTUNO (4)
DUE + CINQUE + CINQUE + NOVE = VENTUNO (3)
TRE + TRE + DODICI + DODICI = TRENTA (3)
TRE + QUATTRO + SETTE + SETTE = VENTUNO (3)
CINQUE + DIECI + DIECI + QUINDICI = QUARANTA (5)
SEI + SEI + OTTO + DIECI = TRENTA (3)
2(CINQUE) + 3(OTTO) + 2(UNDICI) + 2(DODICI) = OTTANTA (5)
10(OTTO) + 2(DIECI) = CENTO (4)
9(DUE) + 2(SEI) + 2(QUINDICI) = SESSANTA (3)
4(CINQUE) + 10(SEI) = OTTANTA (3)
92(UNO) + 4(DUE) = CENTO (5)

In Spanish there are nine doubly true (single solution) alphametics with sums up to 100 where there are no repeated words.

SEIS + CATORCE + SETENTA = NOVENTA (5)
DOS + ONCE + TRECE + CATORCE + TREINTA = SETENTA (4)
TRES + SEIS + SIETE + CATORCE + SESENTA = NOVENTA (5)
CUATRO + CINCO + SIETE + CATORCE + TREINTA = SESENTA (5)
DOS + CINCO + SIETE + DOCE + CATORCE + TREINTA = SETENTA (4)
UNO + CUATRO + CINCO + SEIS + ONCE + TRECE + TREINTA = SETENTA (4)
TRES + CINCO + SEIS + OCHO + ONCE + TRECE + CATORCE = SESENTA (3)
TRES + SEIS + SIETE + ONCE + TRECE + VEINTE + TREINTA = NOVENTA (4)
UNO + DOS + TRES + CUATRO + ONCE + DOCE + TRECE + CATORCE = SESENTA (4)

Here are some Spanish ones with repeated words:

DOS + DOS + TRES = SIETE (2)
UNO + UNO + NUEVE + NUEVE = VEINTE (2)
UNO + CINCO + CINCO + NUEVE = VEINTE (4)
TRES + TRES + CATORCE + SETENTA = NOVENTA (5)
CUATRO + CUATRO + NUEVE + TRECE = TREINTA (4)
CINCO + CINCO + TREINTA + TREINTA = SETENTA (3)
OCHO + OCHO + CATORCE + TREINTA = SESENTA (4)
OCHO + DOCE + TREINTA + TREINTA = OCHENTA (4)
DIEZ + DIEZ + VEINTE + VEINTE = SESENTA (2)
UNO + UNO + UNO + SIETE + DIEZ = VEINTE (2)
UNO + UNO + UNO + SIETE + SETENTA = OCHENTA (2)
UNO + UNO + TRES + SEIS + NUEVE = VEINTE (3)
UNO + UNO + CUATRO + CUATRO + OCHENTA = NOVENTA (4)
UNO + DOS + DOS + CINCO + CINCO = QUINCE (2)
UNO + TRES + TRES + SEIS + SIETE = VEINTE (2)
UNO + CUATRO + CUATRO + ONCE + SETENTA = NOVENTA (5)
UNO + NUEVE + VEINTE + TREINTA + TREINTA = NOVENTA (4)
DOS + TRES + TRES + TRES + NUEVE = VEINTE (3)
DOS + TRES + TRES + CINCO + SIETE = VEINTE (4)
CUATRO + CUATRO + CUATRO + CUATRO + CUATRO = VEINTE (3)
CUATRO + CUATRO + CINCO + SEIS + ONCE = TREINTA (4)
OCHO + ONCE + ONCE + TREINTA + TREINTA = NOVENTA (3)
DIEZ + DIEZ + VEINTE + VEINTE + TREINTA = NOVENTA (4)
DIEZ + VEINTE + VEINTE + VEINTE + VEINTE = NOVENTA (3)
DOCE + DOCE + DOCE + CATORCE + TREINTA = OCHENTA (4)
TRECE + TRECE + CATORCE + VEINTE + VEINTE = OCHENTA (5)
8(DOS) + CATORCE + SESENTA = NOVENTA (4)
8(CUATRO) + 2(CATORCE) = SESENTA (4)
2(DOS) + 14(TRES) + CINCO + 2(SEIS) + 3(OCHO) + TRECE = CIENTO (5)

Here are links to several other web pages on the net that deal with these puzzles.

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Copyright 1998, 1999, 2000 by Truman Collins
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Most recent update: November 10, 2013
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