## Parity and Modular Arithmetic at Bath Time

I am not an expert in early childhood education, so when I was asked to give a presentation at the National Head Start Conference, I had to find a way to establish credibility. I told the audience, “Though I’m not trained as a pre-school teacher, I currently have a pre-school classroom with two students.” I then showed a picture of my twin sons.

Did it establish credibility? I’m not sure. But it was enough to encourage a mother of twins to share some advice with me.

“The best thing we ever did,” she said, “was use even and odd days. One kid got to choose on even days, the other kid got to choose on odd days.” This struck me as sheer brilliance. At the time, my wife and I had a series charts to keep track of whose turn it was: one for who got the Mickey Mouse plate at dinner; one for who got to sit on the passenger side in the car; one for who got to go first when we played Chutes and Ladders; and another fifteen or so for other minutia. It was driving me batty, so this suggestion was a game-changer.

When I returned home from the conference, we immediately implemented this system. We explained to the boys that Alex would get to choose things on odd days, and Eli would get to choose things on even days. (The selection wasn’t arbitrary. Eli would get even days since both Eli and even started with an e.)

Eli said, “But Alex will get to choose two days in a row, on the 31st and the 1st.”

Good point. We decided that the 31st would be mommy and daddy’s day to choose (and in leap years, we’d also claim February 29).

This system worked well, even at bath time, when both boys wanted to sit near the front of the tub to have access to the spigots. Since they took baths every third day, the odd/even system was just fine. Until recently.

About a week ago, we decided to do baths every second day. For this scheme, the odd/even system had three fatal flaws:

• Because of months with an even number of days, there would be strings of up to 31 consecutive baths where the same child would have the choice. (For instance, if baths occurred on even dates in September, they’d also occur on even dates in October; but then in November and December, all baths would occur on odd dates.)
• In any given year, Alex would get the choice between two and five times more than Eli would. (It depends on whether it’s a leap year or not, and on whether the sequence started on an odd or even day in January. But in every case, the system unfairly benefitted the child who receives the choice on odd days.)
• Allowing the 31st to be mommy and daddy’s day to choose doesn’t fully solve the problem. Plus, it smacks of favoritism when a parent chooses one child over the other (for anything).

Uh-oh. I feared that a new chart would be created, and we’d be returned to the bleakness we knew before the even/odd system had been implemented.

Luckily, I was able to devise a new system, and in the process I taught Alex and Eli about modular arithmetic. As a family, we created the chart shown below. The columns indicate those numbers that are congruent to 1, 2, 3, and 0 modulo 4. As shown, Alex would get the choice on days congruent to 1 or 2 mod 4, and Eli would get the choice on days congruent to 3 or 0 mod 4.

But as you can see, this system still unfairly benefits Alex. The solution? Alex does not get the choice if a bath occurs on the 30th of a 31-day month. On those days, I suggested that a coin toss would be used to determine who gets the choice. “If it’s heads, Alex gets the choice; if it’s tails, Eli gets the choice,” I said.

Both boys seemed uncomfortable with this. For some reason, they inherently distrust coin tosses.

So we agreed that we would roll a die instead. “If the roll is even, Alex gets the choice; if it’s odd, Eli gets the choice,” I suggested.

“No,” said Eli. “I get even.”

Such is life in a house with mathematical twins. Everything is a debate. I’m just thankful that it’s a debate about numbers and not about eating broccoli.

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