Triangular Numbers: EYPHKA

The maths teacher was at his wits’ end. To get some respite, he set the class a task:

Add up the first one hundred numbers.

That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later dubbed the Prince of Mathematicians.

Gauss had recognized a pattern. If the numbers from 1 to 100 are written in a row, and again in a second row but in reverse order, the corresponding entries in the two rows add up to 101. With 100 entries in each row, this gives a grand total of 10,100. And since each number occurs twice, the original sum is half this total, or 5050:

Sum1to100Gauss had computed the hundredth triangular number. For any n, the sum of the first n numbers can be arranged in an equilateral triangle. The first few triangular numbers are 1, 3, 6, 10 and 15. For example,10=1+2+3+4. We see this number in the formation of pins in ten-pin bowling. And the reds on a snooker table are set up in a triangle of 15 balls. 

Bowling-PinsFollowing Gauss’s strategy, we can calculate the sum of the first n numbers, or the n-th triangular number, to be T(n)=n(n+1)/2.

The tetractys, a pattern of ten points arranged in a triangle, like bowling pins, was revered as a mystical symbol by the Pythagoreans. Diophantus (c. 210-290 AD), who lived some 700 years later than Pythagoras, discovered an interesting property of triangular numbers: if any such number is multiplied by 8 and augmented by 1, the result is a perfect square, that is, a number multiplied by itself. For example, 8xT(4)+1 = 8×10+1 = 81, which is the square of nine.

Triangular numbers have many other interesting properties. If we add two successive triangular numbers, we always get a perfect square. This can be shown by algebra or by geometry. The geometric argument is illustrated by the figure below, which shows that the two triangular numbers T(4)=10 and T(5)=15 combine to give a square of side 5.

Sum of triangular numbers T(4)=10 and T(5)=15 is a perfect square, 25.

Sum of triangular numbers T(4)=10 and T(5)=15 is a perfect square, 25  [Image: Creative Commons].

Gauss surprised his teacher with his rapid solution of a problem involving triangular numbers. But a more remarkable discovery is found in the mathematical diary that he kept for many years. The entry reads simply

EYPHKA: num = Δ + Δ + Δ

Announced with Archimedes’ cry of discovery, it signals that Gauss had found that every positive number can be expressed as the sum of (at most) three triangular numbers, a result now known as Gauss’s Eureka Theorem.

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