Counting Cattle, Candles, Doves and More

The Tosafos explain how to calculate the total oxen offered for the Musafin of Succos:

תנו רבנן פירשתי מנחה וקבעתי בכלי אחד של עשרונים ואיני יודע מה פירשתי יביא מנחה של ששים עשרונים דברי חכמים

רבי אומר יביא מנחות של עשרונים מאחד ועד ששים שהן אלף ושמונה מאות ושלשים1

שהן אלף ושמונה מאות ושלשים. כיצד קח בידך מאחד ועד ששים וצרף תחילתן לסופן עד האמצע כגון אחד וששים הם ס”א שנים ונ”ט הם ס”א ושלש ונ”ח הם ס”א כן תמנה עד שלשים דשלשים ושלשים ואחד נמי הם ס”א ויעלה לך שלשים פעמים ס”א וכן נוכל למנות פרים דחג דעולין לשבעים כיצד ז’ וי”ג הם עשרים וכן ח’ וי”ב הם עשרים וכן ט’ וי”א הם כ’ וי’ הרי שבעים:2

As Gil Student has noted, this procedure was apocryphally independently discovered about half a millenium later by the young Gauss:

How Many Candles?

How many candles do we light on Chanukah? There are two ways to calculate this. One is to add up every number from 2 (1 + a shamash) through 9 (8 + a shamash) = 44. The other is to recognize that this is an arithmetic series and use the simple formula: 1/2 * n * (A1 + An) = 1/2 * 8 * (2 + 9) = 44.

Now calculate how many candles you would have to light if Chanukah was 15 days or 30 days (135 and 495).

Legend has it that the basis of this formula was figured out by an elementary school student named Carl Friedrich Gauss, when his teacher told the class to add up all the numbers from 1 to 100 and he was able to do it in a few seconds …

R. Mordechai Marcus once told me that there is a Tosafos, I don’t remember where but perhaps in Bekhoros [Menachos 106a s.v. she-hein elef], that offers a similar formula.

(inspired by an e-mail from Dr. Yitzchak Levine)

And about a half a millenium before the Tosafists, the basic method had already appeared in the Propositiones (see previous post):

XLII. propositio de scala habente gradus centum.

Est scala una habens gradus c. In primo gradu sedebat columba una; in secundo duae; in tertio tres; in quarto iiii; in quinto v. Sic in omni gradu usque ad centesimum. Dicat, qui potest, quot columbae in totum fuerunt?

42. proposition concerning the ladder having 100 steps.

There is a ladder which has 100 steps. One dove sat on the first step, two doves on the second, three on the third, four on the fourth, five on the fifth, and so on up to the hundredth step. Let him say, he who can, How many doves were there in all?

Solutio.

Numerabitur autem sic: a primo gradu in quo una sedet, tolle illam, et junge ad illas xcviiii, quae nonagesimo [nono] gradu consistunt, et erunt c. Sic secundum ad nonagesimum octavum et invenies similiter c. Sic per singulos gradus, unum de superioribus gradibus, et alium de inferioribus, hoc ordine conjunge, et reperies semper in binis gradibus c. Quinquagesimus autem gradus solus et absolutus est, non habens parem; similiter et centesimus solus remanebit. Junge ergo omnes et invenies columbas vl.

Solution.

There will be as many as follows: Take the dove sitting on the first step and add to it the 99 doves sitting on the 99th step, thus getting 100. Do the same with the second and 98th steps and you shall likewise get 100. By combining all the steps in this order, that is, one of the higher steps with one of the lower, you shall always get 100. The 50th step, however, is alone and without a match; likewise, the 100th stair is alone. Add them all and you will find 5050 doves.3

As Peter J. Burkholder comments:

It has been related that when Gauss (1777-1855) was a young student, his mathematics teacher one day instructed the class to add the numbers one through 100. No sooner had the assignment been made than Gauss somehow magically produced the correct figure of 5050. How had he done it?

The key to the problem is to realize that by adding corresponding low and high figures, a simple multiplication problem unfolds. Thus, 1+100=101; 2+99=101; 3+98=101;…;49+52=101; 50+51=101. It is manifest from this that one need only multiply the constant sum, 101, by 50, the number of sums. In this way, the correct response of 5050 is obtained.

Alcuin’s ladder problem (42) shows that this concept was already known by the ninth century: …

We can see that with only slight modification, the above-described concept was in place almost a thousand years before Gauss dazzled his schoolteacher. Perhaps the young Gauss wasn’t so clever after all!

  1. מנחות קו. – קשר []
  2. תוספות שם ד”ה שהן אלף []
  3. From Peter J. Burkholder, HOST: An Electronic Bulletin for the History and Philosophy of Science and Technology, 1, #2 (Spring/Summer; June 1993) – link. []
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2 Responses to Counting Cattle, Candles, Doves and More

  1. CS says:

    Wonderful post! I love this blog!

  2. Pingback: Faulhaber’s Formula Of the Figurate Numbers | בין דין לדין

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