Numeric partitions | The game of science


Alfred Young.
Alfred Young.

We wondered last week how you can distribute a cake (or anything else) among three people in such a way that all three accept the distribution as good. In the case of two people it is very simple, and it is also a situation that often occurs in real life, when, for example, two children have to share a chocolate bar or a sandwich: one makes the partition and the other chooses, so the first will take good care to ensure that the two parts are equal. But in the case of three people – whom we will call A, B and C – the question becomes quite complicated. One of the possible solutions (it is not unique, and I invite my sagacious readers to look for others) is the following:

A divides the cake into three parts, B chooses two and the one that remains is for A. If B considers that the two parts he has chosen are equivalent, he gives C the choice between the two and keeps the remainder (although C also You have the option to choose the part of A). If B considers that the two pieces he has chosen are not equal, he cuts a part of one of them to make them equal before offering them to C; in this case there is a residual chunk that the three can be distributed in the same way: one of them (it does not have to be A again) divides it into three parts, etc. Theoretically, the process could repeat itself ad infinitum (or until it reached the molecular level); but in practice it is usually not necessary to go beyond the first step, unless the tripartition of A is clearly inequitable.

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Regarding the division of an obtuse triangle into acute angles, the unnecessary condition that people often impose on themselves when trying to solve it is that all the acute angles have all their vertices on the sides of the obtuse angle, and in this way it is impossible; but, as can be seen in the drawing sent by Enol Ferre, the partition is possible by coinciding several vertices of the acute angles at an interior point of the obtuse angle. Is this division into seven acute angles minimal? (Note, incidentally, the “psychological” parallelism with the famous problem of the nine points to be joined with four rectilinear lines).

Partition of a natural number

And since we have talked about dividing and distributing, it is necessary to mention the mathematical concept of partitioning a natural number, which consists of decomposing it into the sum of other natural numbers (that is, integers and positive).

Partitions is one of those fascinating mathematical topics that, from an extremely simple approach, whose understanding and first developments are within the reach of anyone, opens up a field of unlimited possibilities and innumerable applications.

Let’s see the partitions of the first natural numbers:

The 1 cannot be decomposed into addends, so it only has one partition (the number itself is considered one of its partitions).

The 2 can be decomposed into addends in only one way: 2 = 1 + 1 and, therefore, it has two partitions.

The 3 has three partitions 3 = 2 + 1 = 1 + 1 + 1.

The 4 has five partitions: 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1.

The 5 has seven partitions …

The British mathematician Alfred Young (1873-1940) devised the diagrams named after him to visualize the partitions:

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I invite my astute readers to interpret Young’s diagrams, to relate them to another topic discussed a few months ago in this section, to construct the sequence of the number of partitions of successive natural numbers, and to draw the pertinent conclusions.

Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn maths’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.

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George Holan

George Holan is chief editor at Plainsmen Post and has articles published in many notable publications in the last decade.

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