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

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.

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  • 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.
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  • 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.

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    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.

Seko

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

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  • 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.
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  • 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

Real Mathematics: Numbers #3

Negative integers are a “game changer” when they sneak into our lives. Everything we learn about calculation gets a little bit more complex with the introduction of negative integers. Now we have rules that say crazy things such as “multiplication of two negative integers gives you a positive one”.  How come two negatives make a positive? Is there a sensible explanation?

Q: (-2).(-3) = ?

Algebraic Method: It is being taught that when two negative integers are being multiplied, absolute values 2 and 3 are being done and a plus sign comes to the top of the calculation. This gives +6 or 6 as a result. Almost all of the students learn how to do this calculation but most of them have no idea why it is being done like this.

There are numerous examples to explain the reason. I prefer giving a specific geometric explanation as an example.

Geometric Method: We teach that integers can be shown on an infinite straight line what we call “number line”.  Middle of this line is assigned to the number zero. Left of zero is for negative integers as right side is for positives.

While teaching multiplication with negative integers, you could imagine a line that is perpendicular to the original number line. Upwards would be assigned to the positive integers as negative integers go to downwards.

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Rules

Assume that we have to multiply two numbers.

  1. Sign of the first number tells us which way we are facing: Upwards or downwards. Positive would mean “look up” as negative means “look down”.
  2. Sign of the second number tells us the way we are taking our steps: We could either go forward (which means the number is positive) or go backwards (meaning that the number is negative). You could imagine that backward steps are like how Michael Jackson was “moonwalking”.
  3. Value of the first number tells us how many steps are taken in each time.
  4. Value of the second number tells us how many times those steps are going to be taken.

Let’s solve our original problem with the geometric method: What is (-2).(-3) ?

  • First number is negative: We are facing downwards.
  • Second number is negative: We are taking our steps backwards.
  • Value of first number is 2: We are taking 2 steps in each time.
  • Value of the second number is 3: We are taking those steps 3 times.

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In the end, we are facing down, moving backwards (like Michael Jackson’ moonwalk), 2 steps at a time and 3 times. Our arrival would be +6 as shown on the graphic.

One wonders…

  1. Try to prove that the algebraic method is true. (Hint: Start with assuming that the multiplying two negatives won’t make a positive.)
  2. Try to find another example for geometric method from real life.

M. Serkan Kalaycıoğlu