A good deal

Number Theory Level 3

A shopkeeper sells 1 toffee for 1 Rs. and gives an amazing exchange offer that you can buy 1 toffee by 3 empty toffee wrappers. If you have 1729 Rs., then at maximum how many toffees can you grab from the shopkeeper ?


The answer is 2593.

Ankit Raj by Ankit Raj , 21, India

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2 solutions

Falak Dhruve
Jun 2, 2015

1729 Rs. = 1729 toffees

1729 toffees = 1729 wrappers = 576 toffees + 1 ex1 wrapper

576 toffees = 576 wrappers = 192 toffees

192 toffees = 192 wrappers = 64 toffees

64 toffees = 64 wrappers = 21 toffees + 1 ex2 wrapper

21 toffees = 21 wrappers = 7 toffees

7 toffees = 7 wrappers = 2 penultimate toffees + 1 ex3 wrapper

1 ex1 wrapper + 1 ex2 wrapper + 1 ex3 wrapper = 3 wrappers = 1 ex toffee

1 ex toffee + 2 penultimate toffees = 3 toffees

3 toffees = 3 wrappers = 1 last toffee

Total toffees (sum of all the toffees in bold):

1729 + 576 + 192 + 64 + 21 + 7 + 3 + 1 = 2593 toffees

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Thank you Pi Han HAHAHA!!!!

Noel Lo - 6 years ago

Used this method but missed out something at the end.

Noel Lo - 6 years ago

You start of with 1729 toffees Let n be the number of times you go to the shop to get n candies Then you have:

1729 - 3n + n =1729-2n

1729/2 = 864.5

therefore the no. of candies = 1729 + 864 = 2593

NOTE: this works only if the starting number (1729) is odd. think about it! ;)

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