300 | The game of science

This is the 300th installment of The game of science, which is a good pretext to dedicate it to this singular number. Or, more than singular, “abundant” or “excessive”, which is how numbers that are less than the sum of their divisors without including the number itself are called, or what is the same, their double is less than the sum of all its divisors including the number itself. The divisors of 300 are:

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300

Their sum, 868, is greater than 2 x 300 = 600, so 300 is abundant, and their “abundance” is 868 – 600 = 268.

The abundant numbers are quite abundant: among the first 100 natural numbers there are 22 abundant:

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100

In view of this list, it would appear that all the abundant numbers are even. Is that so? Can you find an odd abundant number that disproves it?

As an anecdote, it should be noted that, regardless of mathematics, the number 300 is “abundant” in another sense: the famous 300 of Leonidas were so widespread because in reality there were about 6,000 (300 full-blooded Spartan Hopilites and several thousand more warriors). ).

300 is also one of those polygonal numbers that we have been dealing with in previous weeks; but which one specifically?

And it is also the sum of two twin primes (two consecutive odd primes): 300 = 149 + 151, and, more difficult still, it is the sum of ten consecutive primes, what are they?

Finally, 300 is a Harshad number (“great joy” in Sanskrit), which means that it is divisible by the sum of its digits (300 is divisible by 3). Among the first 100 natural numbers there are 33 Harshad numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100

Obviously, in any base the one-digit numbers are Harshad numbers.

Tessellation

A problem from the past week has remained pending: How much glue savings does it mean to form a surface with hexagonal tiles glued together at the edges relative to an equivalent surface formed by squares of the same area? Following an unwritten rule of the section, I will not give the solution until some reader is encouraged to solve it; but it is a good pretext to talk about tessellation.

The best known tessellations are the regular ones, studied by Archimedes as early as the 3rd century BC. C. As we have seen on more than one occasion, there are only three regular polygons that can tile the plane: the equilateral triangle, the square, and the regular hexagon.

Semi-regular tessellations, formed by two or more regular polygons, but not arranged in any way, but always repeating the same pattern (in the figure, a semiregular tessellation formed by hexagons regular and equilateral triangles). This formula may seem like a lot of play, but there aren’t many different irregular tessellations. How many?

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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