Summary

How to solve a Summary puzzle: A tutorial for solving cryptarithms and alphametics

This tutorial will help get you started solving Summary. Summary is a number game that thinks it's a word game. A new puzzle ran daily from Jan. 1, 2024 to Aug. 12, 2024, and now you can solve all 225 puzzles here. But because you're here, you may want to do this tutorial first!

What kind of puzzle is this?

Summary is an example of a verbal arithmetic problem, also known as a cryptarithm or alphametic. In short, you need to figure out what digit each letter stands for so that you can create a valid math problem.

The most famous one (famous being a relative term here) is one hundred years old(!) and was published in Strand Magazine by Henry Dudeney. Try it here:
Send + More = Money

Getting the hang of it can be tricky at first, but once you figure it out, it's pretty fun (if we do say so ourselves). This tutorial will help explain some key principles of the game by doing a full walk through of one.

What's the quick version of the tutorial? The summary of Summary, if you will?

The rules say, "Each letter is a unique number. Can you make it all add up?" In a few more words, that means: You are trying to create a valid math problem. Every letter represents a digit from 0-9, and it's the same digit every time it appears. So if there are multiple "A"s in the puzzle, and you found an "A" that has to be a 1, "A" will be 1 everywhere. You'll know you've solved the full puzzle when the top number plus the middle number equal the last number. And remember: numbers can carry! Also: there is only ONE valid solution to any Summary puzzle, and the numbers cannot start with 0.

Still a little unclear? Read on again.

Can I do the tutorial now?

Yes, yes! Enough intro material, we're doing it now.

This is written with a lot of detail, as though we are sitting next to you explaining it (because we've done this many times with friends who may or may not have wanted us to). If you're in a rush, just read the bold parts.

We'll use yummy + tummy = muffin if you want to play along here. (It's just screenshots, not real puzzles, on this page).

1. A very detailed explanation of the rules

When the directions say, "Each letter is a unique number," that means that each individual letter represents a number, and each time the letter appears, it is always the same number. So in this case, M will always be the same number. Same for Y, U and F when they repeat.

When the directions say, "Can you make it all add up?" this means that when you fill in a unique number for each letter, you should end up with an addition problem where the first two numbers add up to the third number. The solution is a valid math problem. So what you see below is a valid solution for AAA + BBB = CCC, because 111 + 222 = 333.

Though a note about the AAA + BBB = CCC example. This would never be a real Summary puzzle, because there can be multiple solutions. For example, 222 + 333 = 555 or 222 + 111 = 333 (and many others) would also work. All summary puzzles have one single solution. There is only one number that can work with each letter.

And one other rule while we're at it. No numbers can start with 0. So in the YUMMY + TUMMY + MUFFIN example, Y, T and M cannot be 0.

2. Find a place to start

The way to start each puzzle is to find a way in. And by that we mean, find a letter that you can figure out, without any other information. (In harder puzzles, you sometimes can't always solve a number immediately, but you can narrow it down. We'll get to that later.) For now, let's start with an easy one.

This puzzle is an easier one, so it has an automatic way in, with a pattern that many puzzles have. When the answer word ("MUFFIN") is longer than the two addends ("YUMMY" and "TUMMY") that means that the leftmost letter of the answer word has to be a 1.

Let's think about why that is. Imagine you had this puzzle, A + A = BC. (Again, not a real Summary puzzle because it has multiple solutions.)

We know that A + A has to equal something 10 or greater, since there are two digits in the answer. So A has to be 5 or more. And it has to carry into the next column. If A were 5, B would be 1. If A were 6, B would be 1. Same for 7, 8 or 9. Even if this isn't totally clear to you why this is true, you can just remember: when the last word is longer than the first two words, the left-most letter has to be 1.

3. Carrying is key

Let's get back to that yummy muffin.

After you've filled in a gimme letter like M, the next step is to see if there are any other letters you can fill based on your new information. By the way, this is essentially the TL;DR of game play: Solve a letter, and then use the new information from that to solve subsequent ones.

That square of Ms is an interesting situation! If you've made it this far, you know that 1 + 1 = 2. But, wait? If M + M = F and M + M = I and I and F can't be the same, how is this possible????

Carrying! This might be the most important trick of the game. As you may recall from around second grade, when you sum up two numbers, columns of numbers that sum up to more than 10 will carry into the next column. So if another column of numbers was carrying into one of the M + M, it's possible that F or I could equal 3. In fact, that is what has to happen here. So we know that between F and I, one has to be 2 and one has to be 3. Which? Think about it for a second before reading on.

Thought about it? F has to be 2 and I has to be 3. Why is that? We know that 1 + 1 cannot carry. It is too small. So nothing will carry into the F column, and so F has to be 2. Then we know I has to be 3! That also tells us something about Y, which we'll get to in a moment.

4. More Carrying Considerations

Now we can tackle our next letter. Let's do U. You do U. What does it have to be?

6! This is another important point of the game. If you see 2 at the bottom, the two numbers above it can add up to 2, or... they can add up to 12. You can often determine based on surrounding letters and numbers which it should be. In this case, we know it has to be 6 because 1 is already taken.

5. Thinking about a few variables at once.

We're almost ready to take it home. But we just have a few more things to do. And this part of the puzzle often makes up the bulk of harder puzzles: looking at multiple variables at once. Let's look at what remains. Y + T = 6 and Y + Y = N. It looks like Y + T = 16 because it has to carry into the left-most column. AND remember that U + U = F is carrying in. So we really know that Y + T have to add up to 15, not 16.

Now, let's also remember what's happening in the right of the puzzle, and also which numbers we have left.

We know a few things about this right column. We know that N has to be even. Why's that? Because two of the same number added up are always even. (There are lots of these qualities of numbers you start to remember that help in this game. This is a common one!) We also know that Y + Y has to carry into the next column to make 1 + 1 = 3.

So let's put this all together and test the numbers we have that remain. This systematic testing is often a part of solving the puzzle (though you could also do it more algebraicly on pen and paper if you'd like).

Of the numbers that remain, we know that N can only be 0, 4, 8. Let's try them.

If N is 0, then Y must be 5. But if we do that then there is no single digit that would work to add up to 15. So 0 is out for N. Let's try 4. That means that Y has to be 7.

So is there anything left that works for T? Yes! 8!

And that is how you do it. There are many, many other strategies for solving these puzzles but this walkthrough covered enough to help you start to figure out more.

6. Practice!

The best way to get better at Summary is to solve more puzzles. We've got 225 of them here. After a few, we know you'll be a pro.

7. A Few Other Advanced Strategies to Keep in Mind

If you don't know any number, start by narrowing with number properties

Sometimes you don't know for sure what any letter is at first, but you can often limit your options down for certain letters, and that is the best place to start. Take SPLISH + SPLASH = BATHES, for example. (And you can try it here.)

You'd first want to notice that S appears several times. And we know a few things about it. From the left-most column, you know that it has to be 1, 2, 3 or 4, because it can't carry. And from the right-most column, you know that it has to be even, because it is the sum of two of the same number with nothing carrying in. So that limits you to 2 or 4, and you can pick one and go from there. You'll quickly be able to derive H and E. You can eliminate an option when you discover that something is creating an impossible situation, and then try another option.

Learn to spot patterns

The pattern with the left-most 1 in the answer is a common one. There are other patterns that you'll start to notice as you do more puzzles. You'll find that the digits 0, 1 and 9 behave a little differently than other digits, making them easier to spot. Can you figure out where 0, 1, and 9 have to go in this puzzle? You can figure it out right off the bat without any additional numbers. (Try the puzzle here.)

A few other questions we've heard from folks, that maybe you, too, have.

Do I need a pen and paper to solve this?

Some players have interesting solving techniques that use a pen and paper. Others don't. It's up to you! We personally find it harder to solve with pen and paper, but that's just us.

I could just program a solver for this.

Yep, you can. If you want to, you should! We did too! That is a fun challenge you can do once, if you know how to program. Solving this with your human mind is a fun challenge you can do many, many times.

What kind of math skills do I need to solve this?

Addition is a must. Some basic algebra also. You don't actually need to write out algebraic equations to solve this (and we think purely taking that approach to solving will actually make life difficult for you), but some basic conceptual algebraic thinking is your friend here. Like, the sort you likely covered in late elementary school or early middle school. In fact, a huge inspiration for creating this was Louis Sachar's Sideways Arithmetic From Wayside School, which is apparently suitable for 9-12 year olds (though also a challenge for those much older).

So is this suitable for kids?

Yes! Summary is great for all math puzzlers, young and old. We know an 8-year-old who is a fan! And we kept that in mind when we designed the puzzles: we didn't include any words in the puzzle we wouldn't want him to see.

How did you make these?

😉

Wait, I have a different question!

Maybe we can answer it! Send us an email at auftup@gmail.com

Okay, you've got this now!

Happy solving!