# digitális matek

### Raymond Smullyan died at the age of 97

„Why should I worry about dying? It’s not going to happen in my lifetime!” quote from Raymond Smullyan, This Book Needs No Title (1986)

In early February, 2017, the mathematical community lost one of its beloved member. Raymond Merrill Smullyan (May 25, 1919 – February 6, 2017) was an American mathematician, concert pianist, logician, Taoist philosopher, and magician.

As well as studying maths and composing chess problems and puzzles, he also learned how to do magic tricks – becoming an accomplished magician. He showed musical talent, winning a gold medal in a piano competition when he was aged 12.

Smullyan is the author of many books on recreational mathematics and recreational logic. Over his career, he wrote numerous research articles on mathematical logic, set theory and Eastern philosophy, while also publishing famous logic puzzles such as „Lady or Tiger„.

### Three logic puzzles. Can you solve them?

1. How Much?
Suppose you and I have the same amount of money. How much must I give you so that you have ten dollars more than I?
2. Large Birds and Small Birds
A certain pet shop sells large birds an d small birds; each large bird fetches twice the price of a small one. A lady came in and purchased 5 large birds and 3 small ones. If, instead, she had bought 3 large birds and 5 small birds, she would have spent \$20 less. What is the price of each bird?
3. The Case of John
This case involved a criminal investigation of identical twins. It was known that at least one of them never told the truth, but it was not known which. One of the twins was named John, and he had committed a crime. (John was not necessarily the one who always lied.) The purpose of the investigation was to find out which one was John.
„Are you John?” the judge asked the first twin.
„Yes, I am, ” was the reply.
„Are you John?” the judge asked the second twin.
The second twin then answered either yes or no, and the judge then knew which one was John. Was John the first twin or the second?

### Solutions to the logic puzzles

1. Solution to puzzle 1:
A common wrong answer is \$10. Now, suppose we each had, say, \$50. If I gave you \$10, you would then have \$60 and I would have only \$40; hence you would have \$20 more than I, rather than \$10. The correct answer is \$5.
2. Solution to puzzle 2:
Since each large bird is worth two small birds, then five large birds are worth ten small birds. Hence five large birds plus three small birds are worth thirteen small birds. On the other hand, three large birds plus five small birds are worth eleven small birds. So the difference between buying five large and three small birds or buying three large and five small birds is the same as the difference between buying thirteen small birds and buying eleven small birds, which is two small birds. We know that the difference is \$20. So two small birds are worth \$20, which means one small bird is worth \$10.
Let us check: A small bird is worth \$10, and a large bird \$20. Therefore, the lady’s bill for five large and three small birds was \$130. Had she bought three large and five small birds, she would have spent \$110, which is indeed \$20 less.
3. Solution to puzzle 3:
If the second twin had also answered yes, the judge obviously could not have known which one was John; hence the second one must have answered no. This means that either both twins told the truth or both lied. But they couldn’t have both told the truth, because it is given that at least one of them always lies. Therefore, both lied, which means that the second twin is John. (It cannot be decided which of the two always lies.)