![]() Similarly "<<" is not possible since A's "<" means it sends its water to a non-existent city on its left. So A discharges (V) and B passes water to the left (<" since this means A sends its water to B and B sends its own to A, so both are using the same pipe and this is not allowed. or else city B can send its sewage to (the left to) A, which treats it with its own dirty water and discharges (V) the cleaned water into the river.or else city A can send its sewage along the pipe (to the right) to B for treatment and discharge, denoted ">V".So A's action is denoted V as is B's and we write "VV" each treat their own sewage and each discharges clean water into the river.Let's represent a city discharging water into the river as "V" (a downwards flow), passing water onto its neighbours as ">" (to the next city on its right) or else ") and list the cities symbols in order. at least one city must discharge the cleaned water into the river.every city must have its water treated somewhere and.or send its own dirty water, plus any from the upstream neighbour, to the downstream neighbouring city's plant, if the pipe is not being used.or send its own dirty water, plus any from its downstream neighbour, along to the upstream neighbouring city's treatment plant (provided that city is not already using the pipe to send it's dirty water downstream).either process any water it may receive from one neighbouring city, together with its own dirty water, discharging the cleaned-up water into the river. ![]() So each city has its own treatment plant by the river and also a pipe to its neighbouring city upstream and a pipe to the next city downstream along the riverside.Īt each city's treatment plant there are three choices: It is more efficient to have treatment plants running at maximum capacity and less-used ones switched off for a period. Water Treatment Plants puzzle Cities along a river discharge cleaned-up water from sewage treatment plants. Auluck in On some new types of partitions associated with Generalized Ferrers graphs in Proceedings of the Cambridge Philosophical Society, 47 (1951), pages 679-686 (examples45 and 46).Ġ, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987. The first Pennies puzzle above was mentioned by F. References Richard K Guy, The Second Strong Law of Small Numbers in The Mathematics Magazine, Vol. P(n): 1 2 5 13 ? n : 1 2 3 4 5 Which one of these two Pennies puzzles is the forgery (it does not continue with a pattern of Fibonacci numbers after some point) and which one genuinely always has Fibonacci numbers of arrangements? There is just 1 pattern with one penny,Īre the P(n) numbers the alternate Fibonacci numbers: i : 0 1 2 3 4 5 6 7 8 Fib(i): 1 1 2 3 5 8 13 21 34. each penny except ones on the bottom row touches two pennies on the row below.each penny must touch the next in its row.One of these puzzles is a fraud, a Fibonacci forgery.So which is the real Fibonacci puzzle?Arrange pennies in rows under these two conditions: The puzzle here is that only one of these two puzzles involves the Fibonacci number series! The other puzzle does not butjust begins with a few of the Fibonacci numbersand then becomes something different. ![]() Each involves arranging pennies (coins) in rows. Pennies for your thoughts - Part 1 Here are two puzzles which are identical - but we count the solutions in twodifferent ways.
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