In the tiny market square just off the Street of the Coppersmiths in Renaissance Pisa, you could find everything from Arabian spices to local cabbages. Maria was looking for wildfowl. She carried a pouch containing 30 identical silver coins, with instructions from the Contessa to spend them all, and to come back with exactly 30 birds. "And I need at least one duck," the Contessa had added.
Maria expected her task to be easy. But when she reached the fowl merchant's stall, she got a shock. "Two coins for a duck?" she cried in horror. "They were half that price last week!"
"It was a special offer, mistress," said the fowl merchant. "It no longer applies. Today, I have pheasants at three coins each, ducks at two coins each, and sparrows at the bargain rate of two birds for one coin. Other kinds of fowl have I not."
"I must spend all 30 coins and buy exactly thirty birds."
"You could buy six pheasants and 24 sparrows."
"No, I must have at least one duck."
"A pretty problem," said the merchant, stroking his beard in puzzlement.
How many birds of each kind should Maria buy?
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The answer
Maria could buy ten ducks and 20 sparrows, or alternatively, 22 sparrows, five ducks, and three pheasants.
To reach the latter solution, begin with the fact that sparrows cost half a coin. As Maria must spend 30 coins, the number of sparrows bought must be even. Start with 30 sparrows and keep reducing that number by two, until you find a possible combination of ducks and pheasants. So, if she buys 28 sparrows then she has 16 coins left, but she has to buy exactly two more birds, which can only cost her a maximum of six coins. Similarly, if she buys 26 sparrows then she has too many coins left over. If she buys 24 sparrows she has 18 coins left and needs six other birds—but the only way to use all the coins is to buy six pheasants. If she buys 22 sparrows she has 19 coins left and needs eight other birds. This can be achieved with three pheasants and five ducks.
The winner is Alan Bowman from Bracknell
