I’m a math teacher so of course one of my most favorite parts of the Seder is the song Echad Mi Yodea, “Who Knows One?” As you probably know, the song counts the numbers from one to 13. The song connects the numbers with topics from the Torah. The song has an interesting pattern. It’s not just one single count. Rather, the first stanza is a count to one, the second stanza is a count to two, etc. That is, each new stanza includes the previous count followed by the next number.

Mathematicians have a word for this pattern. The stanzas are called triangular numbers. That’s strange. How can a number be a triangle? It works like this. Imagine a set of dots organized in rows. The first row is a single dot. The second row is two dots right below the top dot. Then comes three dots below the second row of two dots. Each new row has one more dot than the row above it. Can you visualize this process as a triangle?

Now let’s add up the dots according to the number of rows. If there is only one row, the count is 1. If there are two rows, then the total number of dots is three, that is, 1+2 = 3. If there are three rows, the total count is 6 since 1+2+3 = 6. The numbers 1, 3, and 6 are called “triangular numbers” since they are each the number of dots forming a triangle. The number 1 is the first triangular number, 3 is the second, 6 is the third, and so forth.

You might think that it is difficult to calculate a big triangular number. How can we calculate the 20th triangular number? Don’t we need to do the big addition problem: 1+2 +… + 19 + 20? That’s a lot of additions! There is actually an easier way. I’m going to write the addition problem again and then its reverse on the next line:

1 +2 +… + 19 + 20

20 + 19 +… + 2 + 1

These two sums have the same answer. Only the order of the addition changed. Now add the columns separately. Do you see how we get 21 each time?

21 + 21 +… + 21 + 21

How does this help us get the 20th triangular number? We don’t know the answer just yet. However, this sum of 21’s is twice the answer. That is, it is a sum of two copies of 1 +… + 20. This addition problem of all 21s is much easier to calculate. It is essentially a multiplication problem. That is, it is 21 x 20. This equals 420. Remember how I said that is twice the sum of the numbers from 1 to 20? This means that the 20th triangular number is 210.

Let’s tie everything back to the seder and the song Echad Mi Yodea. What is the sum of all of the numbers in the 13th stanza?

Solution: We can choose to add all the numbers. But let’s try the trick.

1 + 2 +… + 12 + 13

13 + 12 +… + 2 + 1

If we sum the columns we get:

14 + 14 +… 14 + 14

= 14 x 13 = 182

Therefore, when we divide 182 by 2, we get the sum from 1 to 13. This is 91. Shabbat Shalom.

Ari Blinder is a math educator living in Highland Park. He is the owner of Math for the Masses, an innovative tutoring and consulting company. Please visit www.math4masses.com for fun activities and helpful worksheets. Ari can be reached at ari.m4m@gmail.com. Never think you are “not a math person.” You are very much one already, but no one has shown you how.