Logical Problems Created by Inna Shapiro 2006 Problem

Problem 1 n. In some month three Wednesdays fell on even dates. n. What day of week was the 18 th of that month?

Answer ® The first Wednesday could only be the 2 nd day of the month; the second one, the 16 th, and the third one, the 30 th. Otherwise the month would have more than 31 days. ® That means that the 18 th day of the month was Friday.

Problem 2 7 8 5 5 3 4 31 28 or 29 31 30 n. Guess the rule and fill in the empty cells

Answer 7 8 5 5 3 4 4 6 9 31 January 28 or 29 February 31 March 30 April 31 May 30 June 31 July 31 August 30 September n. Thee left column lists the number of letters in the names of the months: the right one, the number of days in those months.

Problem 3 ® Can you continue the row: n. O; T; T; F; F; S; …

Answer ®One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten, Eleven, Twelve… ® Look at the capital letters.

Problem 4 ® Can you move the knight on a chessboard so that it starts in the lower left corner, ends in the upper right corner and visits each square exactly once?

Answer n. Let’s start from a white cell. After the first move we get to a black cell, after the second one, to a white one and so on. To get to the 64 -th cell we have to perform 63 moves. That means we will end up in a black cell, which is a contradiction

Problem 5 ® One water lily bloomed on a lake on the first day of summer. Every day after that the number of blooming lilies doubled. ® On the 20 th day the entire lake was covered by lilies. ® On what day was one half of the lake covered?

Answer ® Let’s start from the last day. In the previous day the number of lilies was 2 times as small. That means one half of the lake was covered on the 19 th day.

Problem 6 ® Can one fill a 5 x 5 square so that all column sums are positive and all row sums are negative?

Answer ® Let us prove that it is impossible. ® Suppose we want to calculate the total sum. If we try to add cells column by column we get a negative result. But if we try to add cells raw by raw we get a positive result. That is a contradiction.

Problem 7 ® There are three cans that can hold 14 gallons, 9 gallons and 5 gallons of milk, respectively. The first one is full of milk, the other two are empty. ® How can you split the milk into two equal parts? You can only use these three cans.

Answer, Part I. ® Let us use the following system of abbreviation. ® “ 3 1; 2, 8, 4. ” means, “Pour milk from can #3 to can #1. After you do that, the first can will hold 2 gallons of water, the second one, 8 gallons, and the third one, 4 gallons. ”

Answer, Part II. ® Start: 14, 0, 0. ® 1 2; 5, 9, 0. ® 2 3: 5, 4, 5. ® 3 1: 10, 4, 0. ® 2 3: 10, 0, 4. ® 1 2: 1, 9, 4. ® 2 3: 1, 8, 5. ® 3 1: 6, 8, 0 ® 2 3; 6, 3, 5. ® 3 1: 11, 3, 0. ® 2 3: 11, 0, 3. ® 1 2: 2, 9, 3. ® 2 3: 2, 7, 5. ® 3 1: 7, 7, 0.

Problem 8 ® There are six numbers: 1, 2, 3, 4, 5, 6 ® You can add 1 to any two of them simultaneously and repeat this process as many times as you wish. ® Can you make all the numbers equal?

Answer ® No ® 1+2+3+4+5+6=21, this is an odd sum ® When you add 1 to two of the numbers, the sum still remains odd. But if all numbers became equal, the sum of six equal numbers would be even. We got a contradiction.