Cryptarithms: recovering digits from constraints alone

0:00Here is a sum with letters where the digits should be. T plus T plus T, and the answer is U T. You are not told what T is, or what U is. And yet exactly one pair of digits works, and you can find it without trying very much at all. These puzzles look like guessing. They are not. A written sum has a structure. Columns feed each other one way only, never backwards.

0:29Once you know that, you stop searching and start deducing. The digits come out one at a time, each one forced. One rule, and it is the whole game. Inside one puzzle, a letter stands for one digit, and it is that digit everywhere it appears. If T is four here, T is four everywhere. That is all. No rule says different letters must be different digits. Two letters may land on the same value, and nothing here needs them not to.

1:02We do assume one more thing, and it is worth saying out loud. A written number does not begin with a zero. Zero seven is not a two-digit number. It is seven. And do not carry values between puzzles. Each starts fresh; a letter that was five a minute ago means nothing here. Back to the first sum. Three copies of T, giving a two-digit answer. The answer's tens digit is U. Its units digit is T — the same T.

1:37So three copies of T have to come back ending in T. Now. Where would you start? Most people start by trying something. Let us try two. Two and two and two is six. Not two. Dead. Three and three and three is nine. Not three either. Dead. That works. But it is a bad habit — ten digits, and this is the smallest puzzle you will meet. Here is the better thing. It is a single idea.

2:08In a column sum, information travels in one direction. Right to left. The units column takes nothing from anywhere. There is no column to its right. So it can be read on its own, cold, with nothing known. Every other column has something arriving from its right — the carry — so you cannot read it until you have read the ones before. That is the method. Start at the units, because it is the only place you can start.

2:40Settle it. Then the next column has one fewer unknown, and you settle that. Starting anywhere else is not a strategy. It is a guess wearing a hat. The units column of our puzzle. T and T and T, ending in T. Walk the ten digits once — not to search, but to close the question for good. One gives three. Two, six. Three, nine. Four, twelve — ending in two. And so on.

3:10Only two digits survive. Zero, because zero and zero and zero is zero. And five, because five and five and five is fifteen, ending in five. Two candidates, from one column, in one pass. And zero fails: three zeros make zero, and zero is not a two-digit answer. So T is five, the total fifteen, and U is what fifteen carries. One. Second puzzle, and it is a step up.

3:43K two, plus K two, equals H M M. K two is not K times two. It is a two-digit number with K in the tens place and a two in the units. So if K were three, K two would be thirty-two. Same method. Units column first. The units column is two plus two. No letter in it at all. It is four. So M is four, before we knew anything else at all. And that column carries nothing, because four is under ten.

4:18Now the far end. The answer has three digits; the numbers being added have two each. So the answer gained a place, and H is where it gained it. How big can H be? The biggest two-digit number is ninety-nine, and two of those come to a hundred and ninety-eight. You cannot reach two hundred by adding two two-digit numbers. Not close, not possible. So H is one. Not probably one. One.

4:48The answer reads one, four, four, and only K is left. The tens column must end in four and hand on the one H needs. Doubling seven gives fourteen, which does both. K is seven. Seventy-two plus seventy-two is a hundred and forty-four. That word carry is doing a lot of work, so let us pin it down. A carry is what a column hands to the column on its left. The useful thing is how small it is.

5:22Add two digits. The most is nine and nine — eighteen. Plus at most one arriving from the right. Nineteen. So a column adding two numbers hands on one, or nothing. Never two. But our very first puzzle added three numbers, not two. Nine and nine and nine is twenty-seven. Plus two arriving, twenty-nine. That column can hand on two. So how much a column passes depends on how many numbers you add. Say a carry is always one, and you are wrong about the very first puzzle here.

6:02Four more, and one question of each. Which column opens it? Y Y plus Z equals Z O O. Two digits plus one digit, giving three. It gained a place, and the most you can make is ninety-nine plus nine — a hundred and eight. So Z is one, and the rest falls out. B five plus three D equals E D five. This opens at the units: five plus D has to end in five, so D is zero.

6:34Then E is one, because the answer gained a place. Seventy-five plus thirty. K P plus K P equals P R R. The gained place makes P one, so the units column is one plus one, and R is two. Sixty-one plus sixty-one. C one plus C equals one F F. Only C of nine lifts a two-digit number to a hundred. Ninety-one plus nine. One opens at the units and three at the far end. That is the only decision you ever make. Which end has something to tell you.

7:15These have a name. They are called cryptarithms, or alphametics. Cryptarithm because the arithmetic is hidden. Alphametic because the letters spell something. Making one is far harder than solving one: you have to arrange for exactly one answer to exist. Which brings up the thing that separates finishing from stopping. Finding digits that work is not the same as showing they are the only ones. Every puzzle here has exactly one answer, and I know because a computer tried all of them — not because you should, but so I can promise the reasoning missed nothing.

7:57And notice what we never needed. Different letters standing for different digits. That rule exists elsewhere and it is a good rule. But nothing here asked for it, and importing rules a puzzle did not ask for is how you get a wrong answer and no idea why. One last puzzle, and this one does not go quietly. U T plus T A equals T A T. Units column. T plus A has to come back as T.

8:29So A is zero. Forced — and the only thing that column can tell us. One letter down, two to go, and the units has nothing left to say. This is where people start trying values. Do not. Go to the other end instead. The hundreds digit of the answer is T. And nothing is in that column except what arrives from the right. So T is a carry — and a carry from two numbers can only be one.

9:00T is one. Then the tens: U plus one must end in zero and hand on that one, so U is nine. Ninety-one plus ten is a hundred and one — every digit out of a column. Look back at what we actually did. We never tried a thousand combinations. That second puzzle had three letters — a thousand combinations — and we tried none. We read the units column: it handed us a digit. We looked at the answer's length: another.

9:35Every digit we settled made the next column smaller. That is the whole trick — and it is not a trick. It is what column addition is. Digits flow leftwards and never back. A sum has a direction, and reading it that way costs almost nothing. So when one has you stuck, the question is never which digit to try. It is: which column can I read without knowing anything? Find that column, and the puzzle unpicks itself.