July 12, 2024
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Adding up to Chanukah: Menorah Math

How many candles do you light in total for all the eight days of Chanukah?

If you answered 44, you are correct. But how long did it take you to get that number? If you say more than thirty seconds, then you probably sat adding up all the numbers from one to eight, and then added eight shamashot. If you took less than 30 seconds, and if you are smirking right now, you probably knew to use Gauss’s formula. There is a famous story about a young man named Gauss whose teacher wanted to keep his students occupied for a long time. So he told his pupils to add up the numbers from one to 100. He was shocked when young Gauss raised his hand after only a few moments with the correct answer, 5050. Gauss had come up with a formula that made it fast and easy to add long lists of consecutive numbers. He made a list of pairs of numbers—the first number and the last, the second and the second to last, etc.—and found that their sums were all equal. Each pair added up to 101.

100 99 98….

1 2 3…..

Since all pairs added up to the same sum, the number of pairs multiplied by the sum of each pair will give you the total sum of all the numbers.

Lets define n as the highest number. If you look at the pairs, you will see that the sum of each pair will be n+1. Each pair is the sum of the highest number (n) and the lowest number (1.) There will be half as many pairs as numbers. So n/2 is the number of pairs.

Therefore, the total sum of 1 till n can be found by n+1 (n/2)

So Gauss did 101(100/2), and got the correct answer: 5050.

This formula can also be used to find the total number of candles we light on Chanukah. We light one on day one, two on day two… this continues until day eight. So if we use Gauss’s formula, we do 9(8/2), and get 36. Then we add eight shamoshot in order to get 44.

Although this formula was famously discovered by Gauss, it really dates back even earlier to the times of the Gemara. A Gemara in Menachot discusses what happens if a person pledged to bring a Korban Mincha but forgot which specific type he had promised to bring. Since there are Minchos that range in measure from one esaron to 60 esronot, Rebbi says he must bring 60 separate korbanot. These 60 korbanot must each have a number of esronot ranging from one to 60, therefore he ends up bringing the sum of one through 60 in order to fulfill his neder. Rebbi says this sum is 1830 esronot.

The Tosafot Menachot daf 106A comments on how Rebbi arrived at this number. It explains how in order to find the sum of many consecutive numbers, you add the first number to the last, the second to the second to last, etc. The highest number is 60, and the lowest number is 1—so 61 is the sum of every pair. It says that if you multiply this sum (61) by the highest number divided by two (30), you will get your answer. 61 multiplied by 30 is 1830, the total sum of all the numbers from one through thirty. Rebbi used “Gauss’s theory” during the time of the Gemara, more than 1000 years before Gauss was even born in 1777.

“Hafoch ba vahafoch ba dichula ba”—If you delve into the Torah enough, you will find everything.

Rachel Retter is a resident of Bergenfield and a junior at Manhattan High School for Girls.

By Rachel Retter

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