“Here begin the problems to sharpen the young” – this is the beginning of a wonderful text believed to have been written by the Carolingian scholar Alcuin of York. He presents over fifty mathematical puzzles, which will challenge even today’s readers. Here are five of our favourites!
Perhaps the most famous of all of Alcuin’s puzzles is the river-crossing puzzle:
A wolf, a goat and a bunch of cabbages.
A man had to take a wolf, a goat and a bunch of cabbages across a river. The only boat he could find could only take two of them at a time. But he had been ordered to transfer all of these to the other side in good condition. How could this be done.
Like with most of the puzzles he wrote, Alcuin also provides you the answer:
Solution. I would take the goat and leave the wolf and the cabbage. Then I would return and take the wolf across. Having put the wolf on the other side I would take the goat back over. Having left that behind, I would take the cabbage across. I would then row across again, and having picked up the goat take it over once more. By this procedure there would be some healthy rowing, but no lacerating catastrophe.
This puzzle has been reproduced many times, with many variations. For example, you can find many videos about it, including this one that was shown at the 1998 Cannes Film Festival:
In his book, The Liar Paradox and the Towers of Hanoi: The Ten Greatest Math Puzzles of All Time, Marcel Danesi explains the genius of this puzzle:
Not only is it included in virtually all the classic puzzle anthologies, but many mathematical historians consider the idea pattern on which it is constructed to be the key insight, that led, centuries later, to the establishment of a branch of mathematics known as combinatorics, which deals essentially with the structure of arrangements. It attempts to determine how things can be grouped, counted, or organized in some systematic way.
Here are four more of Alcuin’s puzzles:
A basilica is 240 feet long and 120 feet wide. It is paved with paving stones one foot 11 inches long and 12 inches, that is one foot, wide. How many stones are needed?
Solution. It takes 126 paving stones to cover the length of 240 feet, and 120 to cover the width of 120 feet. Multiply 120 by 126 and it makes 15120. That is the number of paving stones required to pave the basilica.
An old man greeting a boy.
An old man greeted a boy as follows: “May you live long – as long again as you have lived so far, and as long again as your age would be then, and then to three times that age; and let God add one year more and you will be 100.” How old was the boy at the time ?
Solution. He was 8 years and three months old at the time. The same again would be 16 years and 6 months; double that makes 33 years, which multiplied by 3 makes 99 years. One added to this makes 100.
An abbot with 12 monks.
An abbot had 12 monks in his monastery. calling his steward he gave him 204 eggs and ordered that he should give equal shares to each monk. Thus he ordered that he should give 85 eggs to the 5 priests, and 68 to the four deacons and 51 to the three readers. How many eggs went to each monk, so that none had too many or too few, but all received equal portions as described above.
Solution. Take the 12th part of 204. This 12th part is 17, so 204 is twelve times 17 or seventeen times 12. Just as eighty-five is five times seventeen, so is sixty-eight four times and fifty-one is three times. Now 5 and 4 and 3 are 12. There are 12 men. Again add 85 and 68 and 51 which is 204. There are 204 eggs. Therefore to each one comes 17 eggs as the twelfth part.
A king’s army.
A king ordered his servant to collect an army from 30 manors, in such way that from each manor he would take the same number of men he had collected up to then. The servant went to the first manor alone; to the second he went with one other; to the next he took three with him. How many were collected from the 30 manors?
Solution. After the first stop there were 2 men; after the second 4; after the third 8, after the fourth 16; after the fifth 32; after the sixth 64; after the seventh 128; after the eighth 256; after the ninth 512; after the tenth 1024; after the eleventh 2048; after the twelfth 4096; after the thirteenth 8192; after the fourteenth 16384; after the fifteenth 32768; after the sixteenth 65536; after the seventeenth 131072; after the eighteenth 262144; after the nineteenth 524288; after the twentieth 1048576; after the twenty-first 2097152; after the twenty-second 4194304; after the twenty-third 8388608; after the twenty-fourth 16777216; after the twenty-fifth 33554432; after the twenty-sixth 67108864; after the twenty-seventh 134217728; after the twenty-eighth 268435456; after the twenty-ninth 536870912; after the thirtieth 1073741824.
The English translation of Alcuin’s entire work was done by John Hadley and David Singmaster as part of the article, “Problems to Sharpen the Young” in The Mathematical Gazette, Vol. 76, No. 475 (1992)
See also Marcel Dansei’s article Alcuin’s River Crossing Puzzles and Common Sense