Check Digit

Design the check around the real error, not an imagined one.

1. Read a list of twenty 16-digit numbers aloud while a partner writes them down.
2. Compare against the originals and classify every mistake.
3. Tally: single wrong digits, swapped neighbours, everything else.

Four steps, all mental arithmetic.

1. Starting from the RIGHT of the number, double every second digit.
2. If a doubled result exceeds 9, add its two digits together.
3. Sum everything.
4. The check digit is whatever makes that total a multiple of 10.

Test the claim rather than accepting it.

1. Take your valid number and change ONE digit. Re-check. It fails.
2. Repeat for every position, one digit at a time.
3. Now SWAP two adjacent digits and re-check.
4. Try the swap 0 and 9 specifically.

Put the family side by side and price each one.

1. Tabulate overhead: parity 1 bit per block, Hamming several bits per block, Luhn one digit per number.
2. Tabulate capability: parity detects one error, Hamming corrects one, Luhn detects the human ones.
3. Ask of each: what happens when the check fails?

Hans Peter Luhn, a researcher at IBM, filed the patent in 1954; US 2,950,048 was granted in 1960 and placed in the public domain, which is a large part of why it became universal. Luhn is an interesting figure — he also did foundational work in information retrieval, and the KWIC index is his.

Where you meet it daily: every payment card number ends in a Luhn check digit, which is why a mistyped card is rejected instantly by the web page rather than after a round trip to the bank. IMEI numbers on phones use it. National identification numbers in several countries use it.

Its relatives are chosen for their own error profiles. The ISBN-10 used weights 1 to 10 and a modulo-11 check, which catches ALL adjacent transpositions including 0-9 — at the cost of sometimes needing the check character X, which is exactly why ISBNs occasionally end in an X. ISBN-13 moved to a modulo-10 scheme to align with the barcode system, deliberately giving up that completeness for compatibility. Each design is a specific answer to "which mistakes must never get through, and what may we pay for that?"

What to take from it: before choosing an error-handling approach, characterise the errors. The catalogue holds several — parity, Hamming, check digits, and the retransmission request that needs no code at all — and the right one follows entirely from who or what is making the mistakes and whether anyone can be asked to try again.