Real Mathematics – Numbers #9


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.

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A different kind of magician.

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.

Equal Sums

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.

One wonders…

How did I decide so quickly?

Hint: Take a look at how many of the numbers are odd or even.

M. Serkan Kalaycıoğlu

Real Mathematics – Algorithm #3

Card Games

I remember vividly; at least once a week I was dragged to family meetings. All I wanted was to stay home and play Duke Nukem. But I had to go to those meetings and keep score of a card game which dads play. There were rumors about my high mathematics grades, and I really wanted to fail mathematics just because I might have escaped this responsibility. And game had absurd scores too: “300 to us, 4250 to them, did you write 20 points to us?”

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Everyone who had experienced campus life knows that living in a dorm meant both misery and lots of fun. Personally my favorites were the times when I sat down with three other guys and played card games for hours. After four years I became addicted to the very thing I despised as a child.

I love playing competitive games but if a game has algorithms on the background then I get addicted to that game. In some card games players develop algorithms in their heads. And in a group of close friends, everyone is aware of his/her opponents’ algorithms. This is why making simultaneous adjustments on your algorithm might win you the game.


Pico is a German card game that shows how mathematics can be used in card games perfectly. Even though Pico is a simple game it has astonishing mathematics that lies behind it.


  • In this multiplayer game there are eleven cards which have 2, 3, 4, 5, 6, 7, 8, 9, 10, 13 and 16 written on them.
  • Cards are shuffled and dealt. Each player gets five cards.
  • The only card left gets turned over so that both players know which cards his/her opponent has.
  • In every hand players select a card simultaneously.
  • The card that has the highest number on it wins the hand unless it is more than twice of the other card.
  • The winning card is placed next to the player who won the hand. The number on that card is the score he/she gets.
  • Losing card goes back to its owner.
  • Game continues until either one of the players holds exactly one card.

    In case there is a deck of cards in your home, you could assign K to 13 and J to 16 so that you can play Pico.


I thought about writing a new game which I called Seko:


  • Seko is a multiplayer game just like Pico. There are numbers from 2 to 50. Each player selects six numbers at once. Players roll a dice in order to determine who starts selecting first.
  • Twelve selected numbers are written on a paper.
  • Out of these twelve numbers, each player selects six numbers but this time one by one. Again they roll a dice in order to determine who starts selecting first.
  • In the first hand players select a number simultaneously.
  • If the difference of those numbers is odd, bigger number wins. If the difference is even and less than 20, smallest number wins.
  • The winning number is written in front of the winner. Loser gets his/her number back.
  • Game continues until a player has exactly one number left.

One wonders…

Instead of finding the difference in Seko, add the numbers. If the addition is odd, biggest number wins. If addition is even and less than 50, smallest number wins.

Can you find an algorithm to use in every game of Seko?

M. Serkan Kalaycıoğlu