Fibonacci Sequence Riddles: 8 Puzzles With Answers
A Fibonacci sequence riddle is any puzzle whose answer comes from the series 1, 1, 2, 3, 5, 8, 13, 21, where each number is the sum of the two before it. The famous ones are the rabbit riddle and the stair-climbing puzzle, but Fibonacci numbers also hide inside IQ-test number series. Below are eight puzzles, easiest first, each with the answer and the reasoning behind it.
If you can follow a riddle's answer but can't find it yourself, the slip is often pattern spotting or holding steps in your head, and IQ Tester finds out which: Find my weak spots.
What the Fibonacci sequence is, in simple terms
Start with 1 and 1. Add them to get 2. Add the last two numbers again (1 + 2) to get 3. Keep going:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
That's how the Fibonacci sequence works. The rule is simple: next number = the two before it added together. Some versions start with 0, 1 instead of 1, 1. It's the same series with an extra 0 at the front.
The sequence is named after Leonardo of Pisa, later nicknamed Fibonacci, who described it in his 1202 book Liber Abaci using a puzzle about rabbits. Mathematicians in India had written about the same numbers centuries earlier, in the study of poetry rhythms.
Here are the first few Fibonacci numbers, so you can check your answers as you go:
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Number | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 |
The rabbit riddle
Puzzle 1: The original Fibonacci rabbit problem
You start with one newborn pair of rabbits. A pair can't have babies in its first month, but from its second month on, it produces one new pair every month. No rabbits ever die. How many pairs do you have after 12 months?
Answer: 144 pairs.
Why: Each month, the number of pairs equals last month's pairs (everyone is still alive) plus the number of pairs that are old enough to breed, which is the count from two months ago. "This month = last month + the month before" is exactly the Fibonacci rule. Month by month: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.
Real rabbits don't behave like this, of course. The riddle was always a math puzzle, not biology.
Puzzle 2: The rabbit riddle, reversed
Using the same rules, in which month do you first have more than 50 pairs?
Answer: month 10 (55 pairs).
Why: Month 9 has 34 pairs, month 10 has 55. Reading the table in reverse is a good check that you really understand the pattern.
Stair-climbing puzzles
Puzzle 3: The staircase riddle
You climb a staircase taking either 1 or 2 steps at a time. How many different ways can you climb 5 stairs?
Answer: 8 ways.
Why: Think about your very last move. You either arrived from stair 4 (with a 1-step) or from stair 3 (with a 2-step). So the ways to reach stair 5 = ways to reach stair 4 + ways to reach stair 3. That's the Fibonacci rule again.
- 1 stair: 1 way
- 2 stairs: 2 ways (1+1, or 2)
- 3 stairs: 3 ways
- 4 stairs: 5 ways
- 5 stairs: 8 ways
Puzzle 4: The longer staircase
Same rules. How many ways can you climb 10 stairs?
Answer: 89 ways.
Why: Keep the list going: 6 stairs is 13, 7 is 21, 8 is 34, 9 is 55, 10 is 89. The trick is realizing you don't need to list all 89 routes. Once you spot the Fibonacci pattern, you just add.
This "find the rule, then let the rule do the work" idea is the heart of most reasoning puzzles. It shows up again in how to solve number series questions, where a fixed checklist helps you find the rule faster.
Hidden Fibonacci in number series
On tests, Fibonacci rarely announces itself. It hides.
Puzzle 5: Missing number
What comes next? 2, 3, 5, 8, 13, ?
Answer: 21.
Why: Each number is the sum of the two before it (8 + 13). The giveaway is the gaps: 1, 2, 3, 5. The gaps are Fibonacci numbers too.
Puzzle 6: A Fibonacci-style series with new starting numbers
What comes next? 4, 7, 11, 18, 29, ?
Answer: 47.
Why: Same rule, different start. 4 + 7 = 11, 7 + 11 = 18, 11 + 18 = 29, 18 + 29 = 47. Any series where you add the last two terms behaves like Fibonacci, whatever numbers it starts with. (This particular one is a close cousin called the Lucas numbers, shifted along.)
Here's the hard part: when you get a series like this wrong, was it because you never thought to add two terms, or because you tried it and made an arithmetic slip? It's tough to judge your own mistakes. IQ Tester sorts that out in 16 questions, separating pattern spotting from mental arithmetic, and then gives you a daily drill for the one you actually need. Find my weak spots
Harder variants
Puzzle 7: The three-term twist
What comes next? 1, 1, 2, 4, 7, 13, 24, ?
Answer: 44.
Why: Try the Fibonacci rule: 13 + 24 = 37. That's not it, because 7 + 13 isn't 24 either. Try adding the last three: 4 + 7 + 13 = 24. Yes. So next is 7 + 13 + 24 = 44. Puzzle-makers call this the "tribonacci" sequence. When two-term adding fails, try three.
Puzzle 8: The Fibonacci brain teaser
A number series goes 1, 2, 3, 5, 8, ?, 21, 34. One number is missing. Then someone asks: what is the sum of the first 8 numbers in the full list 1, 1, 2, 3, 5, 8, 13, 21?
Answer: 13, and then 54.
Why: The missing number is 5 + 8 = 13. For the sum, there's a neat shortcut: the sum of the first n Fibonacci numbers equals the Fibonacci number two places further along, minus 1. Two places after 21 is 55, so the sum is 55 − 1 = 54. Check it by adding: 1+1+2+3+5+8+13+21 = 54. It works.
Why Fibonacci shows up on IQ tests
Fibonacci puzzles are popular on reasoning tests and in brain-teaser collections because they test a specific skill: noticing that each term depends on more than one earlier term. Most beginner series only look back one step (add 3, multiply by 2). Fibonacci makes you look back two.
That's why the Fibonacci sequence on IQ tests often catches people who are fast at simple series. They check "what do I add?" and the gaps (1, 2, 3, 5, 8) don't look constant, so they give up. The fix is a habit: if the gaps look like the series itself, try adding two terms.
These puzzles also train a useful habit for all reasoning: say the rule out loud before you calculate. "Each one is the last two added" is easier to test than a vague feeling. The same habit helps with letter series questions, where rules are harder to see.
Fibonacci riddles checklist
- Are the gaps growing but not doubling? Try adding the last two terms.
- Does two-term adding miss by a little? Try three terms.
- Is it a counting puzzle (ways to climb, ways to tile, pairs of rabbits)? Ask "how did I get to this step?" and look for "last + the one before."
- Always check your rule against the whole series.
Turn riddle-solving into a daily habit
Fibonacci riddles are fun, but they also show you exactly where your reasoning slips: spotting the rule, holding the numbers in your head, or adding under pressure. IQ Tester finds which one it is in about two minutes, gives you a reasoning tier, and then sets you one tip and one 10-minute drill a day, weighted toward your weak spots.
FAQ
What is the Fibonacci rabbit riddle?
It's a puzzle from Fibonacci's 1202 book Liber Abaci. One pair of rabbits starts breeding from its second month and has one new pair every month, and none ever die. The number of pairs each month follows the Fibonacci sequence, reaching 144 pairs after 12 months.
How does the Fibonacci sequence work?
Each number is the sum of the two numbers before it. Starting from 1 and 1, you get 2, 3, 5, 8, 13, 21 and so on. Some versions start from 0 and 1, which gives the same series with a 0 in front.
What are some Fibonacci numbers examples in puzzles?
The stair-climbing puzzle (ways to climb stairs taking 1 or 2 steps), the rabbit riddle, and number series like 2, 3, 5, 8, 13 are the most common. Tiling puzzles, where you count ways to fill a strip with 1-wide and 2-wide tiles, use the same numbers too.
Are Fibonacci sequence puzzles on IQ tests?
Fibonacci-style number series appear on many IQ-style and aptitude tests because they check whether you can spot a rule that uses two earlier terms. Knowing the first dozen Fibonacci numbers by heart helps you spot them quickly.
What's a good hard Fibonacci puzzle for students?
Try the tribonacci series 1, 1, 2, 4, 7, 13, 24, where each term is the sum of the last three. Or ask how many ways there are to climb 12 stairs taking 1 or 2 steps (the answer is 233). Both reward spotting the rule rather than counting by hand.