How to Solve Brain Teasers: 8 Tools and When to Use Them

To solve a brain teaser, first decide which type it is, then reach for the tool that suits that type: a grid for logic, the gaps for numbers, the letters for words, and one small change at a time for pictures. A brain teaser is solved when you can show that your answer fits every condition and that no other answer does.

This guide is built around tools rather than reading tips. It sets out eight solving tools, shows which ones suit which type of teaser, and works through five puzzles from start to finish. You can try a practice puzzle at the end.

What counts as a brain teaser?

A brain teaser is a short puzzle that can be solved by reasoning alone. Everything you need is on the page, and the answer can be checked once you have it.

People take them up for more than one reason. A 2020 paper on truth-teller and liar puzzles points out that logical puzzles help people learn logical thinking and also serve as recreation ^[1].

Several classic teasers are older than most people expect. Accounts of the 15 sliding tile puzzle place its beginnings in the 1870s ^[3], and the simplest puzzle about finding one light fake coin with a balance scale first appeared in 1945 ^[2].

What are the main types of brain teaser?

Most brain teasers fall into four types. Each type has its own first move, so naming the type is the quickest way to get started.

Some puzzles mix types. A code lock puzzle uses logic conditions about digits, and a picture puzzle may hide a word. When that happens, start with the part that gives you the most to work with.

Which eight tools solve most brain teasers?

Think of these as a toolbox, not a fixed order. Most puzzles need two or three of them.

1. Sort. Name the type using the table above. The type tells you which tool to try first.
2. Sketch. Put the puzzle on paper as a grid, a list, a number line or a quick drawing. A puzzle you can see is easier than one you hold in your head.
3. Size the search. Count how many answers are possible at all. A short list can simply be checked.
4. Split. Break the puzzle into parts or groups, solve the part you can fix for certain, and treat the rest as a new, smaller puzzle.
5. Assume and follow. Pick one possibility, pretend it is true and follow it to the end. If it breaks a condition, strike it off.
6. Work backwards. Start from the goal or the last condition and move towards the start.
7. Change the view. Look at differences instead of numbers, letters instead of words, or one corner instead of the whole shape.
8. Prove it. Show why your answer fits every condition, and why every other option fails.

Which tools suit which type?

Where do these tools come from?

None of them is new. Formal work on puzzles relies on the same moves, which is a good sign that they work.

Size the search is the key to balance scale puzzles. Every weighing ends in one of three ways: one side drops, the other side drops, or the two sides stay level ^[2]. Because of that, a plan with w weighings can tell apart no more than 3 to the power w coins ^[2]. Knowing the limit tells you how to divide the coins.

Assume and follow is the classic tool for truth-teller puzzles. In knights and knaves puzzles, knights always tell the truth and knaves always lie ^[1]. You suppose someone is a knight, trace what follows, and drop the idea as soon as it clashes with what was said. Puzzles of this kind have also been solved by computers through automated reasoning ^[1].

Split is how large sliding tile puzzles are tackled. A paper published on 10 July 2015 sets out a plain method: place the tiles of the top row, then those of the left column, and then deal with the smaller puzzle that remains ^[3].

Prove it matters because puzzle makers aim for a single answer. In the formal study of truth-teller puzzles, only those with exactly one solution earn the label good ^[1].

How do the tools crack a logic teaser?

The puzzle (The Three Fruits): Sara, Lina and Huda each have one fruit: an apple, a date and a fig. Sara's fruit has five letters. Lina's fruit is not a fig. Which fruit does Huda have?

Sketch and size the search. Draw a grid with the three names down the side and the three fruits across the top. There are only six ways to hand out three fruits, so the search is small.

Split. Fix the person you know most about. Of the three fruit names, only apple runs to five letters, so Sara has the apple. Mark that cell and rule out the rest of her row and the apple column.

Prove it. Lina cannot have the fig, so she has the date, and Huda has the fig. Both conditions hold, and of the six possible hand-outs this is the only one that passes.

Answer: the fig.

How do the tools crack a number sequence?

The puzzle (Growing Gaps): What comes next? 2, 6, 12, 20, 30, ?

Change the view. Stop looking at the numbers and look at the gaps between them: 4, 6, 8 and 10. The gaps grow by 2 each time, so the next gap is 12, and 30 plus 12 gives 42.

Prove it. A second view gives the same result. Each term is a number times the next number: 1 times 2, 2 times 3, 3 times 4, 4 times 5 and 5 times 6. The next term is 6 times 7, which is also 42. When two different rules agree, you can trust the answer.

Answer: 42.

How do the tools crack a word teaser?

The puzzle (The Alphabet Number): I am a number. Spell me out in full and my letters are in perfect alphabetical order. What number am I?

Size the search. There are endless numbers, so the first job is to make the list finite. Any number from one hundred upwards contains the word hundred, thousand or a larger word. In hundred, the n comes after the u, which breaks the order. Thousand fails at once, because h comes before t in the alphabet, and million fails because i comes before m. So only numbers below one hundred can work.

Split. Check the short words first. Zero, one, two, three and the other single words all break the order somewhere, except one: forty. Its letters f, o, r, t and y run forwards through the alphabet with no step back.

Prove it. Two-word numbers such as forty-one fail too. Every tens word ends in y, and no word from one to nine starts with a letter that comes after y. A full check of every number below one hundred leaves forty as the only answer.

Answer: forty, the number 40.

How do the tools crack a picture teaser?

The puzzle (The Lake Word): A row of words is reflected in still water, so top and bottom swap. Only one word looks the same in the lake. Which one?

Split. Do not try to flip whole words. Flip one letter at a time. A capital letter looks the same in still water only if it has a level mirror line across its middle. In simple block capitals, B, C, D, E, H, I, K, O and X work this way.

Prove it. Any word with a letter outside that group changes in the water, so it is out. The word CHOICE uses only C, H, O, I and E. Every letter survives the flip, and a top-to-bottom flip keeps the letters in the same order.

Answer: CHOICE.

How do the tools crack a balance scale teaser?

The puzzle: You have nine coins that look the same. One is lighter than the rest. Using a balance scale only twice, how do you find the light coin?

Size the search. With three possible results per weighing, two weighings give 3 times 3, or 9, different result pairs, so no plan can single out more than 9 coins ^[2]. Nine is exactly the limit, so both weighings must make full use of all three results.

Split. Make three groups of three. Weigh one group against another. If one side rises, the light coin is in that group. If the pans balance, it is in the group left off the scale.

Split again. Take the three suspect coins and weigh one against another. The lighter one is the fake. If they balance, the coin left off is the fake.

Prove it. Each of the nine coins leads to its own pair of results, so the plan always finds the light coin in two weighings.

Which mistakes are most common with brain teasers?

• Using one tool for every type. A grid that cracks a logic teaser does little for a folding puzzle. Sort first.
• Adding rules that are not there. If the puzzle does not say the pieces cannot turn, they may turn.
• Trusting a pattern too early. Two or three terms fit many rules. Check your rule against every term you are given.
• Flipping a picture the wrong way. A reflection in water swaps top and bottom; a mirror on a wall swaps left and right.
• Guessing instead of proving. A bare guess teaches nothing for the next puzzle; a list of ruled-out options does.
• Keeping everything in your head. Paper is part of the method. Sketch the grid, the net or the path.

Can you try one on your own?

Here is a practice puzzle for the size the search, assume and follow, and prove it tools. Each condition rules some digits in and others out, so keep a list and strike off what fails. The answer is hidden until you choose to see it.