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"WHERE," ASKS John Winters, of Wallingford, "does one buy negative red flowers?" Unfortunately his question is less about a gap in the market than about a mistake that stems from last week's bouquet puzzle: "The flowers in a patriotic bouquet are all red bar 11, all blue bar 2 and all white bar 5. How many flowers are there of each colour?"

I had originally written "all red bar 11, all blue bar 12, all white bar 15". This had the merit of working, though the numbers weren't exactly aesthetic. So at the last minute I replaced this ugly set of numbers which worked, by a prettier set that didn't.

A mistake is a learning opportunity. So what lessons can be learnt from this Homeric nod? As my old woodwork teacher used to say "measure twice, cut once". I'd have a wooden hat made with that motto poker-worked into the band if I thought it might fit.

But mistakes are of wider didactic use. The bouquet puzzle underlines the fact that puzzles often involve implicit constraints imposed by the world rather than by the maths. Sheep far from nuclear establishments, for example, occur in whole numbers and, this side of the millennial barrier at any rate, have four legs. To point this up I've devised the first question this week to contain no explicit numbers at all. The numbers that will work in the bouquet question have something to do with triangles. Can you see what it is? And while we're on the subject, on what grounds does the following shepherdly declaration fall short? My sheep are all mottled bar 223 (or should that be baa 223?), all black bar 232 and all white bar 322. How many of each are there?

Last week's Question 3 provides a further learning opportunity and no mistake. Such 2-D objects divisible into a number of equal bits having the same shape as the original are usually called reptiles (Geddit?) in the trade.

Can you see how cutting it in half can provide another shape that can be divided into four reptiles? See fig a.

Last week's puzzles

1 In the numbers from 1 to 1,000,000 inclusive, 0 appears least often (488,895 times) and 1 most often (600,001 times). The rest appear 600,000 times each.

2 There are as many palindromic readings on the 999,999-digit mileometer as on the million-digit one. For each reading with X at its centre on the first there is precisely one with XX on the second, and vice versa. So there must be the same number of each. See fig b.

As for ABSENT SIERRA and ARTISAN BEER, they are anagrams of BRAINTEASER(S)

Points to ponder

I have ducks and pigs only. The number of heads multiplied by the number of beaks gives the total number of legs. How many ducks? How many pigs?

comments to: indy@puzzlemaster.co.uk