On Friday we posed the following puzzle based on Christmas sprouts:

Two farmers met in the pub after a Christmas farmers’ market and started discussing the day’s trade.

“I sold all my sprouts” said the first, “every last one of them”.

“That’s good” replied his friend, “how many did you bring?”

“That’s the odd thing, I can’t remember. But I do remember that in the first hour I sold six sevenths of the sprouts that I brought, plus a seventh of a sprout”

“It must have been awkward cutting a sprout . . .”

“Don’t be daft, I don’t cut them! And strangely, the pattern repeated itself. In the second hour I sold six sevenths of the sprouts I had left, plus a seventh of a sprout. Then in the third hour I sold six sevenths of the sprouts I had left, plus a seventh of a sprout. And finally in the fourth hour I sold six sevenths of the sprouts I had left, plus a seventh of a sprout. Then I came to the pub because I had no more sprouts”.

How many sprouts did the farmer bring to market?

The best way to tackle this is to work back from the final hour. The farmer started that hour with x sprouts and sold 6x/7 + 1/7 sprouts, and as he ended with no sprouts this must equal x.

So x - 6x/7 + 1/7 = (x-1)/7 = 0 and x = 1

At the start of the penultimate hour he has x sprouts and (x-1)/7=1 (as he ends the hour with one sprout) and x is eight.

At the start of the second hour he has x sprouts and (x-1)/7=8 (as he ends the hour with eight sprouts) and x is 57.

And at the start of the first hour he has x sprouts and (x-1)/7=57 (as he ends the hour with 57 sprouts) and he started with 400 sprouts.

The general formula for the number of sprouts is the sum of 7^{0} 7^{1} 7^{2} to 7^{n} where n is the number of hours minus one. With the four hours above this is 1+7+49+343 which is 400.

This number grows very quickly and to sell every sprout using this pattern over seven hours, the farmer would need to bring 137,257 sprouts.