Real Mathematics: Numbers #6

Achtung! Following article contains advertisement for a chocolate company even though I won’t get paid.

How to get the most Milkinis?

Assume that

                – Class has 8 students.

                – All the students adore Milkinis and would do anything to have some.

                – There are 6 Milkinis bars in total.

Rules of the game:

                – 3 tables in the whole classroom named A, B and C.

                – Tables get 1,2 and 3 Milkinis bars in order.

                – On every table each student will get equal share of Milkinis bars. (eg. If there are 4 students on table A, those 4 students will get ¼=0,25 Milkinis bars each.)

                – Ambition is to select the table that has more Milkinis outcome for the student. (eg. While selecting, if student has a chance to get more chocolate from table A, he/she will choose that table.)

Q: Imagine you are one of the 8 students. In order to get the most Milkinis bars, what kind of strategy should you use? (On which turn you should select your table.)

Playing

As long as we don’t change the rules or anything, we will be starting the game in the same fashion every single time: First student will select table C, because it will give him/her 3 Milkinis bars which is greater than A’s 1 and B’s 2 bars.

Second student will choose B, so he/she will get 2 bars of Milkinis.

IMG_4919

Third one has to choose table C as he/she will get 3/2=1,5 bars of Milkinis. Now we have 1 student sitting in table B, 2 students sitting in table C. A is still empty.

IMG_4920

Forth student has 3 identical choices. In all three tables student will be getting same amount of Milkinis bar. (1 Milkinis bar.) Let’s assume forth student selects table B.

IMG_4921

Fifth one will get;

1/1=1 bar from table A,

2/3=0,66 from B,

3/3=1 from C. Let’s say the student selects table C.

IMG_4922

Sixth student will select table A as it will get him/her a full bar of Milkinis.

IMG_4923

Seventh will be selecting table C. Now table C has 4 students and 3 Milkinis bars. Each student here gets ¾=0,75 Milkinis.

IMG_4926

Eight student must select table B as each student on table B gets 2/3=0,66 Milkinis.

Result

Our game is finished with 1 student sitting at the table A, 3 at B and 4 at C. This would result that students get 1 bar of Milkinis from A, 0,66 from B and 0,75 from C. Student who goes to table A is the clear winners in the situation, who was the sixth choice.

One wonders…

  1. Is there a spot while selecting that guarantees the most chocolate?
  2. Why did I feel the need of using decimal point as using fractions would give me the same amount?
  3. One Milkinis bar has 4 little parts. Which student(s) would get the least Milkinis? How many parts of Milkinis would it be?

History of Decimal Point

In early math education teachers discuss decimal point right after teaching what fractions are. If they both mean the same thing, why do we teach both of them?

At the first sight it looks like a waste of time to show the same thing with two different notations. But in truth fractions and decimal point are both very useful and critical in math. Using decimal point might seem confusing, although when it comes to comparing two or more numbers, decimal point notation is easier to the eye than fractions are. (eg. Comparing 0,66 with 0,60 is easier and faster than comparing 2/3 and 3/5.) Also it takes less time when you write down huge numbers as decimals.

Fractions have a history of at least 4000 years. Decimal point notation is relatively a baby next to fractions. In his book History of Mathematics David E. Smith mentions a priest named Christopher Clavius (1537-1612). According to Smith, Clavius is the first known person who used decimal point systematically. In a book his book Clavius made a table called “Tabula Sinuum” where he wrote down his astronomical calculations in decimal point notation.

pellos

In 1492, Francesco Pellos wrote in his arithmetic book that 1/10th of 5836943 makes 583694.3 as shown. Although Pellos wrote the first known decimal point notation in his book, historians of mathematics claim that Clavius should be considered as the inventor of decimal point notation.

M. Serkan Kalaycıoğlu

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