There are seven temples. We want to give same number of flowers to the each temple. Before giving flowers to any temple, we wash them. Whenever we wash our flowers, the amount of flowers we have is doubled.

How many flowers do we need at first and at each temple, so that we have given away all of our flowers after giving to the seventh temple ?

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    $\begingroup$ Uhhhh.... what? $\endgroup$ – dfeuer Jul 18 '13 at 6:51
  • $\begingroup$ It seems that you can always start with one (for whatever number of temples). For seven temples, just wash six times... $\endgroup$ – Dennis Gulko Jul 18 '13 at 6:52
  • $\begingroup$ I think the OP may have meant "floor" and not flower, and the problem thus would be: " We must wash (every temple's floor...?) before we get in it, but if we do the floor('s area...?) gets doubled automatically. How much floor we need (to get washed...?) before we get into the seventh temple? " ...If all the temples have the same area the answer "seems" to be straighforward: $\,7\cdot 2=14\,$ times the area of one single temple, but go figure whether the above is what the OP actually meant... $\endgroup$ – DonAntonio Jul 18 '13 at 7:52
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    $\begingroup$ He is talking about flour offerings to idol worship. $\endgroup$ – Mikhail Katz Jul 18 '13 at 8:39

Start with $x_0$ units of flour, where one unit is required for each idol. After washing if you have $2x_0$ units. You use $1$ unit in the first temple, so you have $x_1=2x_0-1=2(x_0-1)+1$ units when you go to the second temple. After washing this you have $2x_1=2^2(x_0-1)+2$ units; you use $1$ unit in the second temple and take $x_2=2x_1-1=2^2(x_0-1)+1$ units to the third temple. Wash it to get $2^3(x_0-1)+2$ units, leave $1$ unit in the third temple, and take $x_3=2x_2-1=2^3(x_0-1)+1$ units to the fourth temple.

It’s not hard to see (and to prove by induction on $k$, if you wish) that $x_k=2^k(x_0-1)+1$ for each $k\ge 0$, where $x_k$ is the amount of flour left after the $k$-th temple. We want to choose $x_0$ so that no flour is left after the seventh temple has been visited. This means choosing $x_0$ so that $x_7=0$, i.e., so that $2^7(x_0-1)+1=0$; this is an easy equation to solve.


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