By P. A. M. DIRAC

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PM5 48 1/23/96, 10:40 AM p r o b l e m Singular product Bobby added together three consecutive integers, then the next three numbers, and multiplied one sum by the other. Could the product be equal to 111,111,111? PM5 49 1/23/96, 10:40 AM s o l u t i o n Singular product No, it can’t. One of the two triples of numbers must contain an even number and two odd ones. Their sum is even, so the product of the two sums must be even. (S. PM5 50 1/23/96, 10:41 AM p r o b l e m Playing with matches A spiral made of 35 matchsticks is wound clockwise.

So why do they fly upward? PM5 57 1/23/96, 10:47 AM s o l u t i o n Burning question Unburned particles (smoke) are lifted by an upward flow of hot air. When the surrounding air cools, the particles begin to drop and eventually settle to the ground. PM5 58 1/23/96, 10:47 AM p r o b l e m Mix and match There are three red and five blue sticks of different lengths lying on the table. The total length of the red sticks is the same as that of the blue ones. Is it possible to cut up the sticks and pair the pieces such that the pieces in each pair will be alike in length but different in color?

If there are two numbers a and b greater than 1 among the given numbers, then, replacing one of them with ab and the other with 1, we’ll retain the product of all the numbers and increase their sum, because the inequality (a – 1)/(b – 1) > 0 implies that ab + 1 > a + b. Thus, the sum will be greatest if one of the numbers is a billion and all the others are equal to 1. (G. PM5 32 1/23/96, 10:33 AM p r o b l e m Optimist vs. pessimist A glass flask of an irregular shape contains a certain amount of liquid.