To make a triangle out of three stickettes you need the combined lengths of any two to exceed the length of the remaining one. Imagine laying the two sticks end to end in a line (fig a). Suppose we snap the longer stick at a point x metres along it. If the metre length plus x is to be greater than what is left of the two-metre stick, the bit snapped off must be more than half a metre along from the left-hand end of the two-metre stick.
By symmetry, the same must apply to the other end: so if the cut lies within the central metre of the long stick, the bits will form a triangle. If not, not. The cut has a 50 per cent chance of being in this part of the stick, so the chances are 1/2.
We could, instead, have used inequalities. If any two pieces are to be together greater than the remaining one, we must ensure that all of the three following conditions apply:
i) 1+x is greater than 2-x
ii) 1+2-x is greater than x and:
iii) x+2-x is greater than 1.
Now iii) is always true (no matter how you slice it, the two bits of the longer stick total two metres, which is longer than the first stick) while i) boils down to "x is greater than 1/2" and ii) boils down to "x is less than 1.5", which means that the break must lie within the central metre of the longer stick if we are to be able to make a triangle with the pieces - which is what we said before.
Maths is a network of inter-relations, so there are generally many ways of viewing a problem and hence many ways of arriving at an answer to it. This applies even more to puzzles, which is what makes them useful as exercises in creative thinking.
Solution to Last Week's Puzzle
A pig and a pog costs pounds 1, and a pog and a pug costs 52p. A pug and a pig costs even less. How much is a pig? A pog? A pug?
The three pair prices behave like the sides of a triangle (See Last Week's Puzzlemaster), so the missing price must together with 52p exceed pounds 1. That means that a pug and a pig costs more than 48p. The sum of the three pair prices must be even and, as pounds 1 and 52p are even, the missing price must be even too. A pug and a pig costs an even number of pence greater than 48p, ie 50p. Therefore, a pig must cost 49p, a pog 51p and a pug 1p.
Points to Ponder
1 Divide the L-shape into a) four and b) nine equal L-shapes similar to the original. (see fig b)
2 Several of you wrote to me to point out that the four-sided replicons featured last week can be found by bisecting the four bits of the half hexagon. Use a similar trick on the L-shape to get another set of four- sided replicons. (see fig c)
3 What if the pig, pog, pug puzzle had been set in 1984?
4 You have four socks in your sock drawer. Every sock you have is either white or black. If you withdraw a pair at random you have a 50 per cent chance of them turning out to be a black pair. What are the chances of your getting a white pair instead?
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