# PUZZLES

1 Down at The Last Chance

Saloon the Hindoo Jar Brothers have placed a number of white balls and a number of black balls– fewer than 20 in total– into an urn. Gullible Gus withdraws a ball and finds it is black. He replaces it. Sheriff Einstein then withdraws a ball and finds it is white. The chances that two such withdrawals should result in a mismatch (a black ball followed by a white; or a white ball followed by a black) are 42%. How many balls must you withdraw to be sure you have at least one of each colour?

2 In this logo design a right-angled triangle fits snugly into a semicircle in the orientation shown. What should the height and the width of the triangle be if its area is to occupy the greatest fraction of the area of the semicircle? Try to spot the answer without going too deeply into the working!

Wordpool

Find the correct definition: SEA PUSS a) captain’s pet b) sea lion c) pirate’s catamite d) dangerous current

Missing Links

Find a word which follows the first word in the clue and precedes the second in each case making a fresh word or phrase. a) fire band b) trip cutters c) telegraph cat d) fencing city e) sweets scotch f) end ache

1 The chances of a black followed by a white are the same as of a white followed by a black: b/(b + w) X w/ (b + w) ; so we are given that 2bw/ (b + w) 2 = 42/100; this boils down to 200bw = 42(b 2 + w 2 + 2bw); that is: 58bw = 21b 2 +21w 2. Divide both sides by w 2 to get: 21B 2 -58B + 21 = 0. A reassuringly symmetrical quadratic with B = b/w. Solving using the formula (or by inspired factoring) we find reassuringly that B = 3/7 or 7/3. Since b + w < 20, b + w = 10, and we have either 3 black and 7 white or 7 black and 3 white. Either way, we need to withdraw 7 + 1 = 8 to be sure of one of each. 2 Feeling lazy we complete the rectangle. When the rectangle is a maximum our triangle will be maximal, too. Then we get lazier and complete the circle and ask ourselves what sort of rectangle fills most of the circle: the answer is a square. So the triangle must have w = 2h. Such exercises of “seeing it” are invaluable; but they can be backed up by other means.

d);

Wordpool Missing Links

a) fire/arm/band; b) trip/wire/ cutters; c) telegraph/pole/cat; d) fencing/scar/city; e) sweets/ hop/scotch; f) end/ear/ache