## Solution to Puzzle #187: Sum as 100

I am back after a break of two weekends in-between during which I moved from Gurgaon to Bangalore. Lets begin with the answer to this puzzle. I received many correct answers – from Anahat Prakash, Deepak Goyal (my brother) and Pratik Poddar. Well done all!

The answer is “not possible”.  Here is the explanation:

There are 40 pairs of numbers between 10-100 that sum to 100 – (10,90), (11,89)….(49,51). That leaves out the 10 two digit numbers 50, 91, 92…..99. Even if we take all of these 10, and one number each from the 40 pairs, we will still end up with only 50 numbers. If we take any more, we will need to take both the numbers from at least one pair, and they will sum to 100.

I also would like to reproduce the answer from Pratik, who gave the same explanation in another interesting way:

The problem can be changed to:

Is it possible to have 50 numbers from [-40, 49] such that sum of no two numbers is 0.
[41,49] would not matter here

So, Is it possible to have 41 numbers from [-40, 40] such that sum of no two numbers is 0.

Not possible!

Hope you all enjoyed the puzzle!

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