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Logic puzzles by Robert A. Kraus:
1. Inheritance Envelopes.
A wealthy man had three sons all of whom were quite good at math and logic. To get a share of his inheiritance each had to correctly determine a positive integer which he had chosen. He told them that the number had four different non-zero decimal digits, in ascending order.
He prepared three sealed envelopes each of which contained a number. The first contained the product of the four digits, the second contained the sum of the squares of the four digits, the third contained the sum of the product of the first two digits and the product of the last two digits, and the envelopes were clearly marked as such. He showed the three envelopes to the three sons and had them each take one at random.
The sons were stationed at three different computers so that they couldn't communicate with one another (but were linked to the father's computer). After one hour they could submit a number or decline. Anyone who submitted a wrong answer would be eliminated and get nothing. If one or more submitted the correct answer they would each receive a share of the inheritance, and the contest would end with the others getting nothing. If no one submitted the correct answer they would be instructed to work on the problem for another hour. The process would repeat as often as necessary.
At the end of the first hour no one had submitted an answer.
At the end of the second hour no one had submitted an answer.
At the end of the third hour no one had submitted an answer.
At the end of the fourth hour all three of them submitted the correct answer!
Can you determine the number? This puzzle has been solved by Rod Bogart ("Clever puzzle"), Juraj Lorinc, John Gowland, Manning Bartlett, and Stephen Kloder.
2. The Council of Numeria
On the island of Numeria each of the natives is one of two types: Truth-Tellers who always tell the truth, or Liars who never tell the truth. The island is governed by a Council of Elders who will only answer questions that have numerical answers. In fact the only answers they give are whole numbers, either zero or positive. Furthermore, they will never give an answer greater than the current number of council members. This number can vary daily, but is never less than 4 or more than 40.
One day three native students, Ann, Bob, and Cal, were given an assignment by their teacher to question the council. They each asked a question, which was answered by every council member. Afterward they reported to their teacher and made the following statements:
(1) Ann: I asked the council how many of them were Truth-Tellers.
(2) Bob: I asked the council how many of them were Liars.
(3) Cal: Those statements are not both true!
(4) Ann: All of the answers I received were different.
(5) Bob: All of the answers I received were different.
(6) Cal: At least two of my answers were different.
(7) Ann: The sum of my answers is a palindrome.
(8) Bob: The sum of my answers is a palindrome.
(9) Cal: The square root of the sum of my answers is not less than the number of council members.
What was the number of council members on that day? This puzzle has been solved by Stephen Kloder, Stephen Wang, Michael Reid, Dan Blum, Chris Lusby Taylor, Claude Chaunier, Juraj Lorinc, Lance Nathan, Martin Fuller, Roger Phillips, and Seth Kleinerman.
3. MATH CLASS
A high school math teacher chose three of his best students to conduct a little experiment. He said, "I have chosen a three-digit number, N, with the first digit not more than the second and the second not more than the third. I also have chosen a function, F(N), which is one of these five functions:
(1) SUM(N) = The Sum of the digits of N.
(2) PROD(N) = The Product of the digits of N.
(3) SSQ(N) = The Sum of the Squares of the digits of N.
(4) SSC(N) = The Sum of the Cubes of the digits of N.
(5) LCM(N) = The Least Common Multiple of the digits of N.
"I then calculated the value, V = F(N), and have written the three items, N, F, V, each on a separate piece of paper and will give one to each of you. You must try to determine the other 2 items not on your paper. You may use your computers, but cannot collaborate. Don't turn in your answers until I ask for them. Don't worry about any unfair disadvantage of which item you get, this is not a competition, only an experiment. Just make your lists and we'll see what happens!"
Here is what happened:
1:00 - Students begin working.
1:20 - Teacher asks if anyone had found the answers. No one had.
1:30 - Teacher asks if anyone had found the answers. No one had.
1:31 - Teacher asks if it would help if he told them if N was odd or even. All 3 say no.
1:40 - Teacher asks if anyone had found the answers. No one had.
1:41 - Teacher asks if it would help if he told them if V was odd or even. All 3 say no.
1:50 - Teacher asks if anyone had found the answers. No one had.
1:51 - Teacher asks if it would help if he told them the sum of N and V. All 3 say no.
2:00 - Teacher asks if anyone had found the answers. All three of them had!