Pigeonhole Perch

Then one morning, when Perch had been at the post office for thirty-two years, something happened that turned Perch into a kind of detective.

A letter went missing.

This is, by post office standards, not unusual. Letters go missing. Most of the time they are eventually found — usually behind a desk, or in the wrong pigeonhole, or tucked into a colleague's stack. The lost-letter procedure was a standard one. Perch had run it many times.

This particular lost letter, however, was unusual — it had been deposited in the morning collection by a customer who swore she had put it into Perch's hands personally. Perch had no memory of receiving it. The customer was certain. The post office searched. The letter did not turn up. The customer was unhappy. The customer's letter contained important information for her sister, who lived in the eastern province. The matter was, in post-office terms, a bit of a thing.

Perch sat down that evening to think about it.

What Perch noticed was this: the post office had two hundred and forty pigeonholes. The morning collection had contained about two thousand seven hundred letters, by the daily log. The pigeonholes had been sorted twice that day. After the second sort, every pigeonhole should have contained either zero or some small number of letters depending on its destination.

Perch counted, by going through the log carefully, the total number of letters that had been delivered that day from those pigeonholes. The number was 2,699.

The morning collection had been 2,700.

There was one missing letter.

Perch thought about this. Two hundred and forty pigeonholes. 2,700 letters sorted in. Average: a little over 11 letters per pigeonhole. But the average is, Perch knew, just the average. Some pigeonholes had more letters than the average. Some had fewer.

The customer's letter had been addressed to the eastern province. Which pigeonhole was that? It was pigeonhole 113. Perch went to the post office that night, opened pigeonhole 113, and looked carefully.

Pigeonhole 113 had thirteen letters in it.

Perch counted them.

There were thirteen. The log said there had been twelve.

Perch held up the thirteenth letter to the light. It was the customer's lost letter. It had been folded inside another letter — pinched between two pages of a larger envelope — and had therefore been counted as one piece of mail instead of two.

Perch had used the pigeonhole principle to find a lost letter.

(The pigeonhole principle, in case you have not yet met it: if you put more items into a set of boxes than there are boxes, at least one box has more than one item. Perch had used a slight extension — if the count of items in a box is one more than the official log says it should be, then there is an extra item hidden in the box. This is, properly speaking, an extension of the principle. Perch developed it on the job.)

The customer was thrilled. The letter reached the sister. The post office sent Perch a formal commendation. Perch put the commendation in a drawer and went back to sorting.