1 · Alcuin, c. 800
The wolf, the goat and the cabbage
A farmer must take a wolf, a goat and a cabbage across a river. The boat holds the farmer and one of them. Left alone together, the wolf eats the goat and the goat eats the cabbage. How does everyone get across, and in how few crossings?
A first hint
Count every trip, there and back. Which one passenger is safe with each of the other two?
A second hint
Sometimes the farmer has to bring something back the way it came.
Show me the answer
Seven crossings. Take the goat over; come back; take the wolf over; bring the goat back; take the cabbage over; come back; take the goat over.
The record and the legend
- Known The puzzle is in the Propositiones, the oldest known Latin collection of maths puzzles. (Alcuin of York (attrib.), Propositiones ad acuendos juvenes, c. 800; English: J. Hadley & D. Singmaster, "Problems to Sharpen the Young", Mathematical Gazette 76 (1992))
- Disputed The collection is credited to Alcuin, Charlemagne's teacher, in the manuscripts, but scholars are not certain he wrote it. Source
- Known Versions with other animals are told in many countries.
2 · Alcuin, c. 800
Three friends and their sisters
Three friends each travel with a sister. The boat holds two. No sister may be with another friend unless her own brother is there too. How do all six cross, and in how few crossings?
A first hint
Start by sending two sisters over together.
A second hint
At some point two friends cross while their sisters wait on the far bank.
Show me the answer
Eleven crossings. The fewest is eleven; one way: two sisters over, one back, two sisters over, one back, two friends over, a friend and his sister back, two friends over, a sister back, two sisters over, one back, two sisters over.
The record and the legend
- Known Alcuin's version is about friends and sisters; later books made it jealous husbands and wives. (Alcuin of York (attrib.), Propositiones ad acuendos juvenes, c. 800; English: J. Hadley & D. Singmaster, "Problems to Sharpen the Young", Mathematical Gazette 76 (1992))
- Disputed The collection is credited to Alcuin, Charlemagne's teacher, in the manuscripts, but scholars are not certain he wrote it. Source
5 · Alcuin, c. 800
The staircase of pigeons
A staircase has 100 steps. One pigeon sits on the first step, two on the second, three on the third, and so on up to 100 on the top step. How many pigeons are there?
A first hint
Pair the bottom step with the step just below the top: how many pigeons do the two hold?
A second hint
Count how many such pairs there are, and which steps are left over.
Show me the answer
5,050. Alcuin pairs 1 with 99, 2 with 98 and so on: 49 pairs of 100, plus the 50th step and the 100th, makes 4,900 + 50 + 100.
The record and the legend
- Known The puzzle and Alcuin's pairing are in the Propositiones. (Alcuin of York (attrib.), Propositiones ad acuendos juvenes, c. 800; English: J. Hadley & D. Singmaster, "Problems to Sharpen the Young", Mathematical Gazette 76 (1992))
- Disputed The collection is credited to Alcuin, Charlemagne's teacher, in the manuscripts, but scholars are not certain he wrote it. Source
- Legend The story that young Carl Friedrich Gauss summed 1 to 100 in seconds at school comes from a memoir written after his death; the numbers in the classroom have grown in the retelling. Source
6 · Alcuin, c. 800
The snail invited to dinner
A swallow invites a snail to dinner one league away. A league is 1,500 paces, a pace is 5 feet, a foot is 12 inches, and the snail goes one inch a day. Taking a year as 365 days, how long is the trip?
A first hint
Turn the league into inches first.
A second hint
Then turn days into years and days.
Show me the answer
246 years and 210 days (90,000 inches at an inch a day).
The record and the legend
- Known This is the first problem in the Propositiones, with these units. (Alcuin of York (attrib.), Propositiones ad acuendos juvenes, c. 800; English: J. Hadley & D. Singmaster, "Problems to Sharpen the Young", Mathematical Gazette 76 (1992))
- Disputed The collection is credited to Alcuin, Charlemagne's teacher, in the manuscripts, but scholars are not certain he wrote it. Source
8 · Fibonacci, 1202
A pair of rabbits
A pair of rabbits is put in a pen. Every month each pair old enough gives birth to a new pair, and a new pair is old enough after one month. Nobody dies. Starting with one pair, as Fibonacci counted it, how many pairs are there at the end of the year?
A first hint
This month's pairs = last month's pairs + the new ones born to the pairs that were already there two months ago.
A second hint
Write the months in a row and add the two before to get the next.
Show me the answer
377 pairs. Fibonacci counts 1, then 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and 377 at the end of the twelfth month.
The record and the legend
- Known The problem is in Fibonacci's Liber Abaci (1202), with the answer 377. (L. E. Sigler, Fibonacci's Liber Abaci (Springer, 2002))
- Known The same sequence was known in India centuries earlier, in counting the rhythms of poetry. Source
- Disputed Real rabbits do not breed like this; Fibonacci's rules are a model, not a description.
10 · The Sunzi Suanjing, c. 300–500
Counting in threes, fives and sevens
There is a number of things. Counted in threes, two are left over. Counted in fives, three are left. Counted in sevens, two are left. What is the smallest number of things?
A first hint
List the numbers that leave 2 when counted in sevens, and test each one.
A second hint
Keep only those that also leave 2 in threes, then check the fives.
Show me the answer
23 (23 = 3 × 7 + 2 = 5 × 4 + 3 = 7 × 3 + 2). After that, add 105 for the next one.
The record and the legend
- Known This is the problem in the Chinese Sunzi Suanjing that gave the "Chinese remainder theorem" its name. Source
- Disputed Nobody knows who Sunzi was or exactly when the book was written.
11 · Euler, 1736
The seven bridges
A town on a river has two islands joined to the banks and to each other by seven bridges: two from each bank to the first island, one from each bank to the second island, and one between the islands. Can you walk across every bridge exactly once?
A first hint
Count how many bridges touch each piece of land.
A second hint
Except where a walk starts or ends, you must leave a place as many times as you arrive.
Show me the answer
No. Every piece of land touches an odd number of bridges, and a walk like this can have at most two such places, where it starts and where it ends.
The record and the legend
- Known Leonhard Euler proved it in 1736 for the bridges of Königsberg; it is counted as the start of graph theory. Source
- Legend That the townspeople spent Sunday walks trying it is a good story, told often; Euler's paper only says he heard of the problem.
12 · Lucas, 1883
The tower of discs
Three pegs. On one, a tower of discs, each smaller than the one below. Move the whole tower to another peg, one disc at a time, never putting a larger disc on a smaller one. With three discs, what is the fewest number of moves?
A first hint
To move the bottom disc, everything above it must be stacked on the third peg.
A second hint
Moving a tower of two takes three moves. Use that twice.
Show me the answer
7 moves. A tower of n discs takes 2ⁿ − 1 moves, so 64 discs take 18,446,744,073,709,551,615.
The record and the legend
- Known Édouard Lucas sold the puzzle in 1883 under the name "N. Claus de Siam", an anagram of his own name. Source
- Legend The temple where priests move 64 golden discs, and the world ends when they finish, was invented to sell the puzzle. Source