By Raymond M. Smullyan
A variety of puzzles facing notice play and common sense, arithmetic and philosophy, that includes Alice and the creatures of Wonderland
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Additional info for Alice in puzzle-land : a Carrollian tale for children under eighty
Thus 4 X 4 + % equals 17, and 44 + 4 + v'4 equals 50. All numbers up to 112 inclusive may be solved, using only the signs for addition, subtraction, multiplication, division, square root, decimal points, and the factorial sign 4! which means I X 2 X 3 X 4, or 24, but 113 is impossible. It is necessary to discover which numbers can be formed with one four, with two fours, and with three fours, and to record these for combination as required. It is the failure to find some of these that leads to so much difficulty.
Multiply, and also add, 9 and 9, I get SI and IS, which contain the same figures. If I multiply and add 2 and 47, I get 94 and 49-the same figures. If I multiply and add 3 and 24, I get the same figures-72 and 27. Can you find two numbers that when multiplied and added will, in this simple manner, produce the same three figures? There are two cases. 112. PALINDROMIC SQUARE NUMBERS This is a curious subject for investigation-the search for square numbers the figures of which read backwards and forwards alike.
Can you find it? 34 Arithmetic & Algebraic Problems 109. THE TWO FOURS I am perpetually receiving inquiries about the old "Four Fours" puzzle. I published it in IS99, but have since found that it first appeared in the first volume of Knowledge (lSSI). It has since been dealt with at some length by various writers. The point is to express all possible whole numbers with four fours (no more and no fewer), using the various arithmetical signs. Thus 4 X 4 + % equals 17, and 44 + 4 + v'4 equals 50. All numbers up to 112 inclusive may be solved, using only the signs for addition, subtraction, multiplication, division, square root, decimal points, and the factorial sign 4!