There are such questions, even though they seem useless (or unnecessary) to others they can help one to get much better at the number theory. One of the main reasons why some people label those mathematics questions as “useless” is that they are actually scared of the question (or of mathematics).
What those people really feel is like the feeling you get whenever you are walking down on an unknown street, city or country. Those people are away from their comfort zone and they would never feel as relaxed as they feel at home as long as they don’t “try”. Defining mathematics questions as unnecessary is in fact a way of expressing the fear of mathematics.
In that case trying and/or striving are essential if you’d like to get better at mathematics. This is how you could find your own methods and accomplish things that will shock others. There is another matter I’d like to point out: If the trick is obvious no one would watch your show second time. Magic is beautiful when people don’t understand what you are doing.
Assume that we have ten different numbers between 1 and 50. Our goal is to divide those ten numbers into two groups of five such that their summations will be equal to each other.
Example 1: My random ten numbers are: 2, 12, 23, 24, 30, 33, 39, 41, 44 and 48.
Goal is to divide these numbers into two groups of five such that their summations are the same.
I managed to do so after a short amount of time:
48+41+33+24+2 = 148 = 44+39+30+23+12
Maybe you think that I choose those numbers on purpose. This is why I asked some of my friends to send me random ten numbers from 1 to 50.
Example 2: Random numbers: 34, 21, 7, 42, 22, 33, 13, 27, 20 and 19.
After a minute or so I found the following:
34+33+13+20+19 = 119 = 21+22+27+42+7
Q: How do I do it? Can you speculate about what kind of method I might be using?
Example 3: I got these numbers from another friend: 3, 9, 13, 19, 21, 27, 36, 33, 39 and 45.
Example 4: And these are the numbers I received from one last friend: 7, 10, 11, 14, 21, 23, 30, 33, 43 and 49.
For the examples 3 and 4, I found out that there can’t be such groups and I came to this conclusion in a matter of second.
How did I decide so quickly?
Hint: Take a look at how many of the numbers are odd or even.
M. Serkan Kalaycıoğlu