## 23 Aug The Flower Puzzle Generalized

This article is a crazy generalization of the Flower puzzle by Ankush Goswami published in Gonit Sora on 12th July 2012. We now suppose that instead of three there are $m$ temples $A_{1}, A_{2},dots ,A_{m }$ and instead of doubling, the flowers increase $n$ times in number instantly. Let the priest come to the temple with $x$ flowers and keep $y$ flowers in each temple. So after keeping $y$ flowers in temple $A_{1}$ the priest has $nx-y$ flowers which instantly become $n^{2}x-ny$ ; after keeping in temple $A_{2}$ number of flowers become $n^{3}x-n^{2}y-ny$ and so after keeping in the $m$ th temple number of flowers become $n^{(m+1)}x-n^{m}y-n^{(m-1)}y-dots -ny$ which should actually be equal to zero !! Thus

$n^{(m+1)}x-n^{m}y-n^{(m-1)}y-dots -ny =0$

$Rightarrow frac{x}{y}=frac{1+n+n^2+n^3+dots +n^{m-1}}{n^m}$

$Rightarrow frac{x}{y}=frac{n^m-1}{n^m(n-1)}$

The above relation gives all possible values of the number of flowers brought by the priest ($x$) and the number of flowers given by him in each temple ($y$) .

Debashish Sharma, JRF, Dept of Mathematics, NIT Silchar.