A puzzle is a problem where the first idea that pops into your head is often wrong. That's what makes it fun. Good puzzle solvers aren't magic. They carry a small toolbox of strategies: plans you can try when you don't know what to do next.
On this page you'll learn five tools, see each one crack a classic puzzle, and then test yourself with a six-question self-guided quiz.
Many of the puzzles here are old favorites. Some have been passed around for generations, and no one knows who made them up. Writers like Martin Gardner, who wrote the "Mathematical Games" column in Scientific American from 1956 to 1981 (with a few later columns until 1986), helped make puzzles like these famous. The mathematician George Pólya wrote a whole book about how to attack problems, How to Solve It (1945). His simple plan: understand the problem, make a plan, carry it out, and look back.
Tool 1
Work backwards
When a puzzle tells you how things end, start at the end and undo the steps one at a time.
The lily pond
Lily pads on a pond double in area every day. On day 30 the pond is completely covered. On which day was it half covered?
See the worked solution
Most people's first guess is day 15, because 15 is half of 30. But the area doubles, not the number of days.
Work backwards from day 30. The pads doubled to get to day 30. So on day 29 they covered half the pond. Answer: day 29.
Look back: half on day 29, doubled once, gives the whole pond on day 30. ✔
Tool 2
Try the worst case
When a puzzle asks how many you need to be sure, picture the unluckiest thing that could happen. If you're safe even then, you're safe every time.
Socks in the dark
Your drawer has 10 black socks and 10 white socks, all mixed up. The lights are out. How many socks must you grab to be sure you have a matching pair?
See the worked solution
Imagine the worst luck. Your first two socks could be one black and one white. No pair yet.
There are only two colors, so your third sock has to match one of them. Answer: 3 socks.
This idea has a name: the pigeonhole principle. If you put more things than boxes (here, 3 socks into 2 colors), at least one box must hold two things.
Tool 3
Start small, find a pattern
If a puzzle feels too big, solve tiny versions first. Then look for a pattern.
The handshake party
Ten people meet at a party, and every person shakes hands once with every other person. How many handshakes are there?
See the worked solution
Try smaller parties:
Handshakes for small parties
People
2
3
4
5
Handshakes
1
3
6
10
Each new person shakes hands with everyone already there. So the jumps are +2, +3, +4, and so on.
Here is a faster way. Each of the 10 people shakes 9 hands, which gives 10 × 9 = 90. But that counts every handshake twice (once for each person in it). So divide by 2: 90 ÷ 2 = 45. Answer: 45 handshakes.
Look back: the rule "people × (people − 1) ÷ 2" matches the table. For 5 people, 5 × 4 ÷ 2 = 10. ✔
Tool 4
Make a table (guess, check, adjust)
A guess isn't cheating if you check it and use what you learn to make a better guess. A table keeps your guesses organized.
Chickens and cows
A farmer counts 10 heads and 28 legs among her chickens and cows. How many of each animal does she have?
See the worked solution
Chickens have 2 legs and cows have 4. Every animal has 1 head, so there are 10 animals.
Guesses for chickens and cows
Chickens
Cows
Legs
10
0
20
8
2
24
6
4
28 ✔
Notice the pattern: each time you swap one chicken for a cow, you add 2 legs. We needed 28 − 20 = 8 more legs, so that's 4 swaps. Answer: 6 chickens and 4 cows.
An assumption is something you believe without checking. Puzzles love to trick fast thinkers, so test your first answer before you trust it.
The bat and the ball
A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?
See the worked solution
The answer that jumps out is 10 cents. Check it: if the ball is $0.10, the bat is $1.10, and together they cost $1.20. That's too much!
Let the ball cost b. The bat costs b + $1.00. Together: b + b + 1.00 = 1.10, so 2b = 0.10 and b = 0.05. Answer: 5 cents.
Look back: ball $0.05 + bat $1.05 = $1.10, and $1.05 is exactly $1.00 more than $0.05. ✔
Psychologist Shane Frederick used this puzzle in a 2005 study of quick versus careful thinking. Many college students in his study gave the 10-cent answer.
How do we know an answer is right?
Notice the ✔ at the end of every solution. That's Pólya's last step: look back. Put your answer back into the puzzle and see if every fact still holds. A checked answer is one you can defend to anyone.
Self-guided quiz: Use your tools
Six puzzles: one for each tool, plus a bonus snail. Pick an answer for each, then check your score. Click "Check my answers" to see how you did and why.
