Set Of Quickies
The following problems are of the "quickie" type in the sense that they are quickly stated and, at least so I believed when I first gave them, not hard to solve if properly approached. Some are joke questions, and others contain booby traps to catch the unwary.
Problem 1: You want to construct a rigid wire skeleton of a oneinch cube by using 12 oneinch wire segments for the cube's 12 edges. These you intend to solder together at the cube's eight corners.
"Why not cut down the number of soldering points," a friend suggests, "by using one or more longer wires that you can bend at sharp right angles at various corners?"
Adopting your friend's suggestion, what is the smallest number of corners where soldering will be necessary to make the cube's skeleton rigid? (Philip G. Smith, Jr.)
Problem 2: An intelligent horse learns arithmetic, algebra, geometry and trigonometry but is unable to understand the Cartesian coordinates of analytic geometry. What proverb does this suggest? (Howard W. Eves, in Mathematical Circles, Vol. 1.)
Problem 3: Your king is on a corner cell of a chessboard and your opponent's knight is on the corner cell diagonally opposite. No other pieces are on the board. The knight moves first. For how many moves can you avoid being checked? (From David L. Silverman's collection of game problems, Your Move.)
Problem 4: Nine heart cards from an ordinary deck are arranged [see Figure 35] to form a magic square so that each row, column and main diagonal has the largest possible constant sum, 27. (Jacks count 11, queens 12, kings 13.) Drop the requirement that each value must be different. Allowing duplicate values, what is the largest constant sum for an order3 magic square that can be formed with nine cards taken from a deck? (M. G.)
Figure 35
Vv 
y 
Aa 
A magic square with nine hearts Problem 5: Make a statement about n that is true for, and only true for, all values of n less than one million. (Leo Moser.) Problem 6: Why would a barber in Geneva rather cut the hair of two Frenchmen than of one German? Problem 7: With a black pencil draw a closed curve of any shape you please. With a red pencil draw a second curve of the same kind on top of the first one, never passing through a previously created intersection. Circle all points where one curve crosses the other [see Figure 36], Prove that the number of such points is even. (M. G.) Problem 8: Place a familiar mathematical symbol between 2 and 3 to express a number greater than 2 and less than 3. Problem 9: A sixstory house (not counting the basement) has stairs of the same length from floor to floor. How many times as high is a climb from the first to the sixth floor as a climb from the first to the third floor? Problem 10: Each of the two equal sides of an isosceles triangle is one unit long. Without using calculus, find the length of the third side that maximizes the triangle's area. Problem 11: What three positive integers have a sum equal to their product? Problem 12: A string, lying on the floor in the pattern shown in Figure 37, is too far away for you to see how it crosses itself at points A, B and C. What is the probability that the string is knotted? (L. H. LongleyCook, Fun with Brain Puzzlers.) Problem 13: If AB, BC, CD and DE are common English words, what familiar word is dcabe? (David L. Silverman, Word Ways, August 1969.) Problem 14: Time, March 7, 1938, reported that one Samuel Isaac Krieger claimed to have found a counterexample to Fermat's unproved last theorem. Krieger announced that it was 1,324" + 731"= 1,961", where n is a certain positive integer greater than 2, and which Krieger refused to disclose. A reporter on The New York Times, said Time, easily proved that Krieger was mistaken. How? Problem 15: What familiar English word begins and ends with und? Problem 16: A man arrives at a random spot several miles from the Pentagon. He looks at the building through binoculars. What is the probability that he will see three of its sides? (F. T. Leahy, Jr.) Problem 17: Change 11030 to a person by adding two straight line segments. Problem 18: A boy and a girl are sitting on the front steps of their commune. "I'm a boy," said the one with black hair. "I'm a girl," said the one with red hair. If at least one of them is lying, who is which? (Adapted from a problem by Martin Hollis, in Tantalizers.) Problem 19: A "superqueen" is a chess queen that also moves like a knight. Place four superqueens on a fivebyfive board so that no piece attacks another. If you solve this, try arranging 10 superqueens on a 10by10 board so that no piece attacks another. Both solutions are unique if rotations and reflections are ignored. (Hilario Fernandez Long.) 
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frances6 years ago
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