How many oranges?

Number Theory Level pending

A fruit seller goes to the Raja to deliver his oranges. On his way, there are 7 gates and 7 guards guarding them. He gives half of his oranges to the 1st guard and gets back one from him. He repeats this for the rest of the guards,too. Finally, when he reaches the Raja, he has 2 oranges left. How many oranges did he originally bring?


The answer is 2.

Rajalakshmi Madhavan by Rajalakshmi Madhavan , 22, India

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2 solutions

Dawar Husain
Dec 26, 2014

Reverse engineer! The number of oranges is operated upon by two operations - Divide by 2 and add of 1, 7 times.

Now, reverse this process for the remaining no. of oranges, ie, Subtract 1 and Multiply by 2

Therefore, no. of oranges = [ ( ( ( ( ( 2 − 1 ) × 2 ) − 1 ) × 2 ) − 1 ) × 2 ] … [(((((2-1)\times 2)-1)\times 2)-1)\times 2] \ldots 7 times.

After each iteration, the result is 2 \boxed{2} . Hence the answer.

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Kumar Saurav
Dec 26, 2014

calculate how many he had before the last gate & continue. with each step one gets 2

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