Earlier today I set you two puzzles by the veteran US puzzle inventor Sam Loyd.
1. The famous hot cross bun puzzle
This familiar nursery rhyme conceals a riddle:
Hot cross buns, hot cross buns,
One a penny, two a penny,
Hot cross buns.
If you have no daughters
Give them to your sons!
Two a penny, three a penny,
Hot cross buns.
I had as many daughters
As I had sons.
So I gave them seven pennies
To buy their hot cross buns.
In other words, hot cross buns come in three kinds: those sold at one a penny, two a penny and three a penny. There are as many boys as girls. The children spend exactly seven pennies on buns.
The puzzle is asking that if every child gets the same number and kinds of buns, how many buns, and of what kinds, does each child receive? (Assume that the buns remain whole.)
Solution: There are three boys and three girls. If you spend 3 pennies on ‘two a penny’ buns, and 4 pennies on ‘three a penny’ buns, you can divide them up so that every child gets a single ‘two-a-penny’ bun, and two ‘three-a-penny’ buns.
The wording of the question reveals that there must be at least two boys (and girls), and thus there are at least 4 children. However, if you assume there are 4 children, there are no ways to divide the buns in the correct way. But if there are 6 children, there is. You must plug the numbers in to see what works and what doesn’t.
User Mark_1023 pointed out BTL the ‘trivial’ solutions in which 1 boy/1 girl, or 7 boys/7girls, buys only two-a-penny buns . However I think Sam Loyd worded the questions to eliminate these possibilities: his phrasing leads the reader to infer that there at least two children of each gender, and that they buy at least two types of bun.
2. A study in eggs.
How many eggs can you put in a 6×6 crate such that no row, column, or diagonal has more than two eggs in it? The first two have already been placed, in opposing corners, as shown above. Can you draw me their positions?
Hint: the answer is more than 10 (including the two already in the crate).
Solution. The maximum possible is 12, since there are six rows and columns. In fact, 12 is possible. Here’s one way:
I hope you enjoyed today’s puzzles. I’ll be back in two weeks.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book.
Today’s puzzles originally appeared in Sam Loyd’s Cyclopedia of Puzzles, published in 1914. They were reprinted in 1960 by Dover Publications in More Mathematical Puzzles of Sam Loyd, edited by Martin Gardner.