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The candles each take one hour to burn completely. Cutting off bits of the candles is forbidden, but the candles are placed on a raft of fork handles so they may be burnt at both ends (e.g. to time $1/2$ hour burn a candle at both ends). You are also allowed to weld n candles to each other and light x ends if $x<2n$ (e.g. $2/3$ hours can be timed by welding $2$ candles and lighting $3$ ends) to get a fraction of $\frac{n}{x}$. 1/16 hours can be timed with four candles by lighting all the candles at one end, and the first candle at both ends. When that candle is burnt light the second candle at the other end. Then light the third at the other end, then the fourth which will take 1/16 hours to finish burning.

Given any rational number of hours, what is the fastest way to determine the minimum number of candles needed?

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How do you get 3 ends by welding 2 candles to each other? Do you mean orthogonally? –  user18325 Dec 20 '11 at 8:19
    
Sounds like "fusible numbers": mathpuzzle.com/fusible.pdf See also this related question: math.stackexchange.com/q/40404/409 –  Blue Dec 20 '11 at 8:32
    
The title says "four candle problem", but the body never mentions the number of candles. If there really are only four candles, you obviously can't measure any (presumably positive) rational number of hours. Could you please clarify that? Also, do you have a physical mechanism for "welding $2$ candles and lighting $3$ ends" in mind, or is this to be taken just as an additional mathematical operation that we're allowed to perform? –  joriki Dec 20 '11 at 13:58
    
I'm reading this as follows: you can measure any rational time $p/q$ with $1 \le q \le 2p$ using $p$ candles by welding them together and lighting $q$ fuses. Presumably you can also sequentially carry out two operations, so if you can measure $t_1$ and $t_2$ with $n_1$ and $n_2$ candles respectively, then you can measure $t_1 + t_2$ (and maybe $|t_1 - t_2|$?) with $n_1 + n_2$ candles. The remaining question is if there are any other "fusibility axioms": can you light some of the fuses of an $n$-candle blob, wait a (measured) time $t$, and then light some more? If so, what happens? –  mjqxxxx Dec 20 '11 at 16:10
    
If $2/3$ can be obtained, it’s not simply fusible numbers: all of them are dyadic. Like others, I really don’t understand the welding mechanism, even as an abstraction, unless you just mean that welding $n$ candles and lighting $k$ ends by fiat produces $n/k$ hours. –  Brian M. Scott Dec 20 '11 at 18:00

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