 ## 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.

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