We know that the second plague in Mitzrayim was frogs. We all know the songs about frogs going everywhere throughout the land. But according to our tradition, it didn’t have to be that way. The pasuk (Shemot 8:2) uses the word “frog” in the singular, and according to the midrash this means there was only one frog at first. According to the midrash, the Egyptians kept hitting it in anger and it split into multiple frogs!

How long did it take for frogs to get everywhere? One possibility is that the population of frogs doubled every minute by so many Egyptians hitting them. That’s a little extreme, though, because after an hour, there would be over 1 billion billion frogs in Egypt! On the other hand, if only a constant number of Egyptians hit the frogs each minute, the number of frogs would grow very slowly. Is there an in-between possibility?

There is. Say there were 9 frogs already and then 7 were hit, then next 9, then 11, then 13, and continuing each time with the next odd number of frogs being hit by the Egyptians. How many frogs would there be after an hour? (Try to find the pattern without adding every number.)

Solution: There would be 3969 frogs after an hour. This is 63 X 63. The pattern for the population growth is the square numbers. For example,

9+7 = 16 = 4 X 4.

16+9 = 25 = 5 X 5,

25+11 = 36 = 6 X 6,

36 + 13 = 49 =7 X 7

This list is offset by 3, because it started with 9 frogs. Therefore, after 60 minutes the number was 63 x 63 = 3969. (If you are curious about why this pattern keeps going on when you add the next odd number, experiment with squares and L-shapes surrounding them,) Shabbat Shalom.

Ari Blinder is a math educator living in Highland Park, New Jersey, and the owner of Math for the Masses, an innovative math tutoring and consulting company. For more information, visit https://www.math4masses.com. Ari can be reached at [email protected].