Every single-digit error in 3141592653, under weights 1…10 mod 11
check-digit is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
What each check-digit scheme catches
No pair of weights modulo ten catches everything
Swap-proof scrambles exist for odd moduli and for the pentagon
The pentagon's ten symmetries, and where their order matters
Damm's table, a Latin square with a zero diagonal
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the integers modulo 2 have a swap-proof map exactly when 2 is odd ×10
- 2363 passes and its swap 2633 fails ×2
- a scheme that weights every position the same catches no transposition ×1
- and every single-digit error too ×1
- Damm catches every single error ×1
- Damm catches every swap of neighbours ×1
- Damm's table is not a group's ×1
- descending weights over a prime modulus catch every transposition ×1
- every row and column holds each digit once ×1
- every single-digit error is caught exactly when the modulus is at least ten and coprime to every weight ×1
- every symmetry can be undone ×1
- exactly one check digit completes a number under Damm ×1
- exactly one check digit completes a number under Luhn ×1
- exactly one check digit completes a number under Verhoeff ×1
- exactly one check digit completes a number under weights 3,1 mod 10 ×1
- no pair of weights modulo ten catches every single error and every swap ×1
- no running value lets a swap through ×1
- the best a pair of weights catching every single error can do on swaps is eight in nine ×1
- the count for eleven is the known 37,851 ×1
- the diagonal is zero ×1
- the error kinds are single, transpose or twin ×1
- the number is between four and fourteen digits ×1
- the number is between six and fourteen digits ×1
- the pentagon's symmetries have swap-proof maps ×1
- the scheme is one of sum-9, sum-10, flat-11, weighted-10, isbn-11 ×1
- the sums modulo ten each miss some swaps ×1
- the table is associative ×1
- Verhoeff catches every single error ×1
- Verhoeff catches every swap of neighbours ×1
- Verhoeff's permutation never lets a swap through ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
Distance is a picture
A message is a corner of a cube and an error is a step along an edge. Everything a code can do is decided by how far apart the corners it uses are — and that is a fact about a drawing.
ComputationTen digits need a symmetry that does not commute
A check digit should catch one wrong digit and two neighbours swapped. Over eleven symbols a weighted sum does both; over the ten decimal digits no weighted sum can, no scheme of any shape built on adding modulo ten can, and the reason is the same as the reason Euler's thirty-six officers cannot be paraded. What works is the ten symmetries of a pentagon, which do not commute.