Thursday, August 7, 2008

Cleaning Up the Leftovers - Solution to Last Parsha Challenge

I am a bit disappointed that after 3 days, not one reader tried to solve this math problem. It leads me to draw one of 2 obvious conclusions:

My readers are not mathematically inclined - (could be, could be not)
I do not really have many readers - (BINGO!)

As you recall, our question from the end of Parshas Balak was:

If there were 625,730 Jews remaining after this mass "execution", how many Jews were there before the execution and how many judges?

This actually requires nothing more than 8th grade (or is it 9th grade?) Algebra.

According to the simple understanding of the Gemara in Sanhedrin 18a, which Rashi seems to follow, the ratio of judges to the general population ran as follows:

For every unit of 1000 Jews there were:
  • 1 Sar Elef
  • 10 Sarei Meah
  • 20 Sarei Chamishim
  • 100 Sarei Assarot

This amounts to 1 + 10 + 20 + 100 = 131 judges within every unit of 1000. In other words, 13.1% of the population were judges. If each judge executed 2 Jews, then the total amount of Jews executed was 26.2% of the population. As such, the remaining 625,730 Jews were 73.8% of the original population of Jews.

So - to arrive at the correct number, we simply have to calculate - 625,730 / .738 = OP (Original population). According to my handy-dandy calculator, this works out to: 625,730 / .738 = 847,872 (we will drop the remainder) Jews.

And so, it comes out like the simple pshat that there were originally 847,872 Jews, of whom 13.1% or 111,071 were judges. They executed two people each or 222,142 Jews. So after 222,142 Jews were eliminated from the original 847,872, we get 847,872 - 222,142 = 625,730. Of these, 24,000 perished in the subsequent plague leaving the number recorded in Parshas Pinchas - 601,730.

Now, let's check that. In 847,872 we should expect to find:

  • 848 Sarei Elef
  • 8,479 Sarei Meah
  • 16,957 Sarei Chamishim
  • 84,787 Sarei Assarot

847 + 8,478 + 16,957 + 84,787 = 111,070 - Not bad!

Now, the identical figures will hold up according to the first answer of Tosafos because the only difference is that the judges were not part of the population count. Regardless, their cumulative number is still 13.1% of the population count and so, to execute 26.2% of the population and remain at 625,730 we must still begin at 847,872.

Tosafos's second answer will only cause a slight change. According to the second answer, each higher ranking judge was the "top gun" from the group he came from, and so, the total number of judges actually equalled the number of its lowest rank- the Sarei Asaros which is one out of 10 (10%) of the population. So if all the judges were 10% then those executed were 20% and the remainig population were 80%. Thus, we calculate 625,730 / .8 = 782,162. Herein, approximately 78,216 were judges and they executed 156,432. Sure enough, 782,162 - 156,432 = 625,730.

Anyway you look at it, the number quoted in Rashi falls short of the mark. Of course I am not comfortable to assume that Rashi was not "on the ball" and much prefer to say that Rashi was not being precise (lo chash l'dakdek) about the population, possibly so as not to complicate the matter and confuse the reader.

What bothers me more is, why is it that I have seen absolutely no commentary take issue with this?

They must be like my readers!

1 comment:

Sammy Davis, Jr. said...

Here's one vote that it's the math. As soon as I saw the problem, my eyes glazed over and I scrolled down to the next item.