Chanukah Candles Mathematics

What is Hanukkah?

Hanukkah is a Jewish festival commemorating the recovery of Jerusalem and subsequent rededication of the Second Temple at the beginning of the Maccabean Revolt against the Seleucid Empire in the 2nd century BCE.

From Wikipedia

How Hanukkah candles are lit?

During Hanukkah festival it’s a custom to light candles on a special type of candelabrum that has eight plus one places for candles. Eight places to light a candle each day and one for an auxiliary candle (shammash in Hebrew) that is intended to light other candles.

During eight days of the holiday a candle is lit in such a way that on first day you light 1 candle, on second day 2 candles and on it goes until on the eighth day 8 candles are lit.

How many are there?

So then the natural question to ask is how many candles do I need to have to be able to light candles for eight days.

The math says sum them up,

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) + 8 = …

And you get the number. Remember that additional 8 candles are there since we need to count the auxiliary candle used each day. Even though this calculation isn’t difficult to make, this may quickly change if you need to sum 1 to 100 candles.

What can you do? Is there a quick way to sum the candles? Indeed there is and if you recall your school math you could remember something about arithmetic progression. It gives you the formula below to count any number of candles or any other objects for that matter:

Sn = (a1 + an) * n / 2,

where Sn – sum of n elements, a1 – first element, an – last element, n – total number of elements.

For n elements the formula is

Sn = (1 + n) * n / 2

So in our particular case of Hanukkah it will be

S8 = (1 + 8) * 8 / 2 = 9 * 4 = 36 + 8 = 44 (again don’t forget auxiliary candles)

While this formula is correct one may wonder how do you derive it? What is the intuition? You always can resort to mathematical induction to prove why it works correctly. We leave it for you as an exercise (a free Book Of Proof by Richard Hammack could be helpful). But there is a visual way to understand how this formula was derived.

On the shoulders of giants

It is a folk legend that when one of the greatest mathematicians Carl Friedrich Gauss attended a school a teacher asked pupils to count numbers from 1 to 100 to calm them down. It says that Gauss came up very quickly with the answer 5050.

And the explanation is that he did this trick.

1, … 100 = 101

2, … 99 = 101

3, … 98 = 101

4, … 97 = 101

… …

97, … 4 = 101

98, … 3 = 101

99, … 2 = 101

100, … 1 = 101

He placed numbers form 1 to 100 in such a way that first sequence was ascending from 1 to 100, and the second was descending from 100 to 1. Then if you look at the sum of each pairs it makes 101. There are 100 such pairs, so when you multiply them you get

101 * 100 = 10100

But we counted each pair twice, so there is a need to divide this number by two.

101 * 100 / 2 = 5050

You may think at this point, well Gauss was a genius so he came up with this explanation, but it still feels like I do not get it fully. And I agree. It is still not fully clear how he came up with this trick. So let’s try to look at it more visually.

Visual aids to the rescue

If you look at candles that are lit on each day of Hanukkah from above we’ll see something like on the figure below.

You’ll see that this figure resembles a lot a right triangle. So it’s good to think about this in this way for a moment.

Now, you may recall that to get an area of a rectangular you multiply its two sides

Area = a * b

Well, rectangular is build out of two right triangles isn’t it? So if you look carefully at the figure below you’ll see what Gauss possibly thought.

You have 8 candles on one side of the rectangular. Then you have 9 candles on the other side of the rectangular. The area of this rectangular is

Area = 8 * 9 = 72,

but since we are only interested in half of the rectangular then let’s divide it by two.

72 / 2 = 36.

And don’t forget about adding 8 auxiliary candles.

So we get

36 + 8 = 44

Let there be light

That’s it folks. Let the light of the candles enlighten us in math.

Take care.


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