During the week, I accumulate about $3 worth of change in my trouser pocket. It mainly serves my kids and their needs for milk money and so forth. Toward the end of the week, I start to spend down the change, counting out large and precise amounts of it at stores. I’ve begun to notice something. In the entire pile, I seldom find a nickel.

Look, here is today’s change, scattered on my desk: 8 quarters, 11 dimes, 8 pennies — and one nickel. That’s absolutely typical.

I began to ask around. Conclusion? Many retail change-makers are such numskulls they do not observe anything. (I did find out that modern cash registers do not actually count coins for the clerk, though the register does calculate the amount of change owed. I had thought that might be a reason for the nickel’s scarcity — that the machines eliminated it.) One day after Christmas, however, I found my local post empty except for the clerks, and I asked the two of them, one man, one woman, whether they had noticed how seldom nickels were used.

“Oh, yeah,” the man said immediately. “We go to the bank, we never get nickels. Nickels last forever.”

What accounts for the scarce appearance of nickels? Numeric necessity? There are 99 coin-change transactions, from one cent to 99. Assuming that all such transactions get executed with the minimum number of required coins, a nickel is necessary in eight, plus four times eight. The eight are 5, 15, 30, 40, 55, 65, 80, and 90 cents. “Four times eight” represents the additional number of transactions represented by the four pennies for odd amounts up to the next five-cent interval: 6, 7, 8, and 9, and 31, 32, 33, and 34, for example. Total, 40. And each of those 40 transactions requires only a single nickel.

By contrast, every coin-change transaction from 25 cents up requires at least one quarter. That’s 75 quarters right there. Plus one more quarter required from 50 cents up, for a total of 125, and one more from 75 cents up, for a total of 150.

Dimes: Every transaction from 10 cents up except those exclusive to quarters or to quarters and nickels requires at least one dime. Those are 10, 15, 20, 35, 40, 45, 60, 65, 70, 85, 90 and 95. Twelve dime transactions, plus four more for each, representing the odd-penny add-ons up to the next five, for a total of 60. Of those 12, three, 20, 45, and 95, take an additional dime. I’ve already accounted for the “plus four pennies” algorithm with the first single-dime run-through, so, out of 99 transactions model, dimes will be required 63 times.

On a strict mathematical basis, change-making requires three and a half times as many quarters as nickels, and not even twice as many dimes as nickels.

But something else must be going on here. My typical pocket sample has 10 times as many dimes as nickels, and eight to 10 times as many quarters. And, as the postal clerk told me, “Nickels last forever.”

Why?

I suspect that clerks often substitute dimes for quarters, probably because they notice their change drawers running short of quarters (it’s the biggest coin and the easiest to see). So you’d get a quarter and three dimes for 55 cents instead of two quarters and a nickel.

Or perhaps retail clerks can’t add. My local small grocery limits cash back from a debit card to $25. The cash registers don’t add that amount, so the clerks have to add it on paper or in their heads, and many simply can’t.

Or maybe many retail clerks, seriously challenged in English, hail from countries where there are no coins denominated in fives. You can learn to say “small black” or “chocolate frosted” quickly, but counting comes hard in a different language.

Any ideas?