Project P063
Check digits
Luhn, ISBN-10 and ISBN-13 / EAN-13 check digits: check a number, add the check digit to one without, and test a valid number against every single wrong digit and every swap of neighbouring digits to see which slips each scheme catches - and which it lets through.
2 modules · 2 recorded sessionstext-menu UI in the terminalupdated 2026-10-14
Every screen below was recorded under CPython. When this page was built, the EML interpreter replayed each session from the same input and printed the same bytes.
About
Three check-digit schemes: Luhn, used on card and phone-equipment numbers, ISBN-10, and ISBN-13 / EAN-13, the barcode number on books and goods. You can check a number, add the check digit to a number without one, or test a valid number against every single wrong digit and every swap of two neighbouring digits, to see which slips the scheme catches and which it lets through. A check digit guards against slips of the hand, not against anyone faking a number.
main.eml- the menu, choosing a scheme, reading digits, and the messageschecks.eml- the three weighted sums, the check digit, and the error test
How each part works:
- Luhn: as in the corpus case
luhn-algorithm, start from the right and double every second digit, taking 9 off a result above 9; the number is valid when the total is a multiple of 10. - ISBN-10: the digits weigh 10, 9, ... down to 1 from the left, and the total must be a multiple of 11. The check digit can need the value 10, which is written X.
- ISBN-13 / EAN-13: the digits weigh 1, 3, 1, 3, ... from the left, and the total must be a multiple of 10.
- The check digit is the one last digit that makes the number valid; since the last digit weighs 1, exactly one does.
- The error test changes the number every possible way that a typing slip would: each digit to each other digit, and each pair of different neighbours swapped. It counts what the check catches. - All three catch every single wrong digit. - Only ISBN-10 also catches every swap, because 11 is prime. - Luhn lets a 09 / 90 swap through: doubled, 0 stays 0 and 9 becomes 18, which counts as 9, so both orders add up to 9. - EAN-13 lets a swap of two digits 5 apart through: with weights 1 and 3, a swap changes the total by twice the difference of the two digits, and twice 5 is 10.
What is checked: menu choices 1 to 4; schemes 1 to 3; digits, with spaces and hyphens skipped. An X is allowed only as an ISBN-10's last digit. The lengths are 2 to 19 digits for Luhn, 10 for ISBN-10 and 13 for EAN-13, each one fewer before the check digit is added. An error test starts from a valid number. An empty answer cancels.
Sessions: sessions/basic.in gives: - the published examples 79927398713 (Luhn), 0-306-40615-2 (ISBN-10) and 978-0-306-40615-7 (ISBN-13), each checked with its sum; - check digits added: X for 155404295, 1 for 400638133393, and 7 for 40903; - error tests: - 409037 under Luhn misses its 09 and 90 swaps; - 4006381333931 under EAN-13 misses the 38 swap; - 0306406152 under ISBN-10 catches every swap.
sessions/bad-input.in gives: - menu choices 0 and x, and scheme 4; - ISBN-10 input 12345, abc, and 15540429X5 with an X that is not last; - 79927398710, which is not valid (its check digit should be 3), checked and then error-tested; - a 5-digit EAN-13 body.
Built on the verified corpus case luhn-algorithm (doubling every second digit from the right, the total a multiple of 10).
Recorded sessions
What the screen shows while someone uses the program. Each typed line appears after its prompt, the way a terminal shows it.
bad-input
interpreter: byte-equal== Check digits ==
Luhn, ISBN-10 and ISBN-13 / EAN-13: the last digit catches most slips of the hand.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 0
Pick a number from 1 to 4.
1) check a number 2) add the check digit 3) error test 4) quit
choice> x
Pick a number from 1 to 4.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 1
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 4
Type 1, 2 or 3.
scheme>
Cancelled.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 1
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 2
digits (spaces and hyphens are fine)> 12345
An ISBN-10 number has 10 digits; that one has 5.
digits (spaces and hyphens are fine)> abc
Use digits only, with an X allowed as the last one.
digits (spaces and hyphens are fine)> 15540429X5
Use digits only, with an X allowed as the last one.
digits (spaces and hyphens are fine)>
Cancelled.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 1
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 1
digits (spaces and hyphens are fine)> 79927398710
79927398710 is not valid: its sum is 67, which leaves 7 over 10. With these first digits the check digit would be 3.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 3
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 1
digits (spaces and hyphens are fine)> 79927398710
Start from a valid number - use 2) to add its check digit.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 2
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 3
digits (spaces and hyphens are fine)> 12345
An ISBN-13 / EAN-13 number has 12 digits before its check digit; that one has 5.
digits (spaces and hyphens are fine)>
Cancelled.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 4
Bye.
What was typed (22 lines)
0
x
1
4
1
2
12345
abc
15540429X5
1
1
79927398710
3
1
79927398710
2
3
12345
4
basic
interpreter: byte-equal== Check digits ==
Luhn, ISBN-10 and ISBN-13 / EAN-13: the last digit catches most slips of the hand.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 1
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 1
digits (spaces and hyphens are fine)> 79927398713
79927398713 is valid: its Luhn sum is 70, a multiple of 10.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 1
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 2
digits (spaces and hyphens are fine)> 0-306-40615-2
0306406152 is valid: its ISBN-10 sum is 132, a multiple of 11.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 1
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 3
digits (spaces and hyphens are fine)> 978-0-306-40615-7
9780306406157 is valid: its ISBN-13 / EAN-13 sum is 100, a multiple of 10.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 2
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 2
digits (spaces and hyphens are fine)> 155404295
The check digit is X: 155404295X.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 2
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 3
digits (spaces and hyphens are fine)> 400638133393
The check digit is 1: 4006381333931.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 2
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 1
digits (spaces and hyphens are fine)> 40903
The check digit is 7: 409037.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 3
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 1
digits (spaces and hyphens are fine)> 409037
One wrong digit: 54 of the 54 ways are caught.
Two neighbouring digits swapped: 3 of the 5 swaps are caught.
not caught: 09 -> 90 at digits 2 and 3
not caught: 90 -> 09 at digits 3 and 4
1) check a number 2) add the check digit 3) error test 4) quit
choice> 3
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 3
digits (spaces and hyphens are fine)> 4006381333931
One wrong digit: 117 of the 117 ways are caught.
Two neighbouring digits swapped: 8 of the 9 swaps are caught.
not caught: 38 -> 83 at digits 5 and 6
1) check a number 2) add the check digit 3) error test 4) quit
choice> 3
1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13
scheme> 2
digits (spaces and hyphens are fine)> 0306406152
One wrong digit: 91 of the 91 ways are caught.
Two neighbouring digits swapped: 9 of the 9 swaps are caught.
1) check a number 2) add the check digit 3) error test 4) quit
choice> 4
Bye.
What was typed (28 lines)
1
1
79927398713
1
2
0-306-40615-2
1
3
978-0-306-40615-7
2
2
155404295
2
3
400638133393
2
1
40903
3
1
409037
3
3
4006381333931
3
2
0306406152
4
Modules
The program as written, entry module first. Each module transpiles to its own Python file, which is what eml project run executes.
main.eml(entry)
eml# P063 check digits: check a number against Luhn, ISBN-10 or ISBN-13 /
# EAN-13, add the check digit to a number without one, and test which
# typing errors each scheme would catch on a given number. A check digit
# guards against slips of the hand, not against anyone faking a number.
import checks
def trim(s):
0 => i
len(s) => j
while i < j and s[i] == " ":
i + 1 => i
while j > i and s[j - 1] == " ":
j - 1 => j
return s[i:j]
def ask_scheme():
" 1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13" ^0
while True:
trim(input("scheme> ")) => answer
if answer == "":
return -1
if answer == "1" or answer == "2" or answer == "3":
return int(answer) - 1
"Type 1, 2 or 3." ^0
def ask_digits(scheme, with_check):
# The number's digits, or [] when the answer is empty.
checks.lengths(scheme) => span
span[0] => low
span[1] => high
if not with_check:
low - 1 => low
high - 1 => high
while True:
trim(input("digits (spaces and hyphens are fine)> ")) => answer
if answer == "":
return []
checks.digits_of(scheme, answer, with_check) => ds
if len(ds) == 0:
if scheme == 1 and with_check:
"Use digits only, with an X allowed as the last one." ^0
else:
"Use digits only." ^0
elif len(ds) < low or len(ds) > high:
if low == high and with_check:
("An " + checks.names()[scheme] + " number has " + str(low) + " digits; that one has " + str(len(ds)) + ".") ^0
elif low == high:
("An " + checks.names()[scheme] + " number has " + str(low) + " digits before its check digit; that one has " + str(len(ds)) + ".") ^0
else:
("Type " + str(low) + " to " + str(high) + " digits; that one has " + str(len(ds)) + ".") ^0
else:
return ds
def check():
ask_scheme() => scheme
if scheme == -1:
"Cancelled." ^0
return
ask_digits(scheme, True) => ds
if len(ds) == 0:
"Cancelled." ^0
return
checks.weighted_total(scheme, ds) => total
checks.modulus(scheme) => m
if checks.valid(scheme, ds):
(checks.as_text(ds) + " is valid: its " + checks.names()[scheme] + " sum is " + str(total) + ", a multiple of " + str(m) + ".") ^0
else:
len(ds) => n
checks.check_digit(scheme, ds[0:n - 1]) => want
(checks.as_text(ds) + " is not valid: its sum is " + str(total) + ", which leaves " + str(total % m) + " over " + str(m) + ". With these first digits the check digit would be " + checks.digit_text(want) + ".") ^0
def complete():
ask_scheme() => scheme
if scheme == -1:
"Cancelled." ^0
return
ask_digits(scheme, False) => body
if len(body) == 0:
"Cancelled." ^0
return
checks.check_digit(scheme, body) => d
("The check digit is " + checks.digit_text(d) + ": " + checks.as_text(body + [d]) + ".") ^0
def test_errors():
ask_scheme() => scheme
if scheme == -1:
"Cancelled." ^0
return
ask_digits(scheme, True) => ds
if len(ds) == 0:
"Cancelled." ^0
return
if not checks.valid(scheme, ds):
"Start from a valid number - use 2) to add its check digit." ^0
return
checks.error_test(scheme, ds) => r
("One wrong digit: " + str(r[0]) + " of the " + str(r[1]) + " ways are caught.") ^0
("Two neighbouring digits swapped: " + str(r[2]) + " of the " + str(r[3]) + " swaps are caught.") ^0
for miss in r[4]:
(" not caught: " + checks.digit_text(miss[1]) + checks.digit_text(miss[2]) + " -> " + checks.digit_text(miss[2]) + checks.digit_text(miss[1]) + " at digits " + str(miss[0] + 1) + " and " + str(miss[0] + 2)) ^0
"== Check digits ==" ^0
"Luhn, ISBN-10 and ISBN-13 / EAN-13: the last digit catches most slips of the hand." ^0
True => running
while running:
"" ^0
"1) check a number 2) add the check digit 3) error test 4) quit" ^0
trim(input("choice> ")) => choice
if choice == "1":
check()
elif choice == "2":
complete()
elif choice == "3":
test_errors()
elif choice == "4":
False => running
else:
"Pick a number from 1 to 4." ^0
"Bye." ^0
Python projection (main.py)
import checks
def trim(s):
i = 0
j = len(s)
while i < j and s[i] == " ":
i = i + 1
while j > i and s[j - 1] == " ":
j = j - 1
return s[i:j]
def ask_scheme():
print(" 1) Luhn 2) ISBN-10 3) ISBN-13 / EAN-13")
while True:
answer = trim(input("scheme> "))
if answer == "":
return -1
if answer == "1" or answer == "2" or answer == "3":
return int(answer) - 1
print("Type 1, 2 or 3.")
def ask_digits(scheme, with_check):
span = checks.lengths(scheme)
low = span[0]
high = span[1]
if not with_check:
low = low - 1
high = high - 1
while True:
answer = trim(input("digits (spaces and hyphens are fine)> "))
if answer == "":
return []
ds = checks.digits_of(scheme, answer, with_check)
if len(ds) == 0:
if scheme == 1 and with_check:
print("Use digits only, with an X allowed as the last one.")
else:
print("Use digits only.")
elif len(ds) < low or len(ds) > high:
if low == high and with_check:
print("An " + checks.names()[scheme] + " number has " + str(low) + " digits; that one has " + str(len(ds)) + ".")
elif low == high:
print("An " + checks.names()[scheme] + " number has " + str(low) + " digits before its check digit; that one has " + str(len(ds)) + ".")
else:
print("Type " + str(low) + " to " + str(high) + " digits; that one has " + str(len(ds)) + ".")
else:
return ds
def check():
scheme = ask_scheme()
if scheme == -1:
print("Cancelled.")
return
ds = ask_digits(scheme, True)
if len(ds) == 0:
print("Cancelled.")
return
total = checks.weighted_total(scheme, ds)
m = checks.modulus(scheme)
if checks.valid(scheme, ds):
print(checks.as_text(ds) + " is valid: its " + checks.names()[scheme] + " sum is " + str(total) + ", a multiple of " + str(m) + ".")
else:
n = len(ds)
want = checks.check_digit(scheme, ds[0:n - 1])
print(checks.as_text(ds) + " is not valid: its sum is " + str(total) + ", which leaves " + str(total % m) + " over " + str(m) + ". With these first digits the check digit would be " + checks.digit_text(want) + ".")
def complete():
scheme = ask_scheme()
if scheme == -1:
print("Cancelled.")
return
body = ask_digits(scheme, False)
if len(body) == 0:
print("Cancelled.")
return
d = checks.check_digit(scheme, body)
print("The check digit is " + checks.digit_text(d) + ": " + checks.as_text(body + [d]) + ".")
def test_errors():
scheme = ask_scheme()
if scheme == -1:
print("Cancelled.")
return
ds = ask_digits(scheme, True)
if len(ds) == 0:
print("Cancelled.")
return
if not checks.valid(scheme, ds):
print("Start from a valid number - use 2) to add its check digit.")
return
r = checks.error_test(scheme, ds)
print("One wrong digit: " + str(r[0]) + " of the " + str(r[1]) + " ways are caught.")
print("Two neighbouring digits swapped: " + str(r[2]) + " of the " + str(r[3]) + " swaps are caught.")
for miss in r[4]:
print(" not caught: " + checks.digit_text(miss[1]) + checks.digit_text(miss[2]) + " -> " + checks.digit_text(miss[2]) + checks.digit_text(miss[1]) + " at digits " + str(miss[0] + 1) + " and " + str(miss[0] + 2))
print("== Check digits ==")
print("Luhn, ISBN-10 and ISBN-13 / EAN-13: the last digit catches most slips of the hand.")
running = True
while running:
print("")
print("1) check a number 2) add the check digit 3) error test 4) quit")
choice = trim(input("choice> "))
if choice == "1":
check()
elif choice == "2":
complete()
elif choice == "3":
test_errors()
elif choice == "4":
running = False
else:
print("Pick a number from 1 to 4.")
print("Bye.")
checks.eml
eml# P063 check digits - three schemes. A number is a list of digits 0 to 9,
# with 10 standing for the X an ISBN-10 may end in. Each scheme adds a last
# digit so that a weighted sum comes out a multiple of 10 (or 11), and a
# typing error usually breaks that.
def names():
return ["Luhn", "ISBN-10", "ISBN-13 / EAN-13"]
def lengths(scheme):
# [shortest, longest] number, check digit included.
if scheme == 0:
return [2, 19]
if scheme == 1:
return [10, 10]
return [13, 13]
def luhn_total(ds):
# The corpus case luhn-algorithm: from the right, every second digit is
# doubled, 9 taken off when that passes 9, and everything added up.
0 => total
len(ds) => n
for i in [0:n - 1]:
ds[n - 1 - i] => d
if i % 2 == 1:
d * 2 => d
if d > 9:
d - 9 => d
total + d => total
return total
def weighted_total(scheme, ds):
if scheme == 0:
return luhn_total(ds)
0 => total
if scheme == 1:
# ISBN-10: weights 10, 9, ..., 1 from the left.
for i in [0:9]:
total + (10 - i) * ds[i] => total
return total
# EAN-13: weights 1, 3, 1, 3, ... from the left.
for i in [0:12]:
1 => w
if i % 2 == 1:
3 => w
total + w * ds[i] => total
return total
def modulus(scheme):
if scheme == 1:
return 11
return 10
def valid(scheme, ds):
return weighted_total(scheme, ds) % modulus(scheme) == 0
def check_digit(scheme, body):
# The last digit that makes body valid: try each, 0 to 9 (and X, 10,
# for ISBN-10). Exactly one works, since the last digit's weight is 1.
10 => top
if scheme == 1:
11 => top
for d in [0:top - 1]:
if valid(scheme, body + [d]):
return d
return -1
def digit_text(d):
if d == 10:
return "X"
return str(d)
def as_text(ds):
"" => s
for d in ds:
s + digit_text(d) => s
return s
def digits_of(scheme, s, with_check):
# The digits of s, spaces and hyphens skipped, or [] when anything else
# is there. An X (either case) is allowed only as an ISBN-10's check
# digit.
[] => out
"" => kept
for ch in s:
if ch != " " and ch != "-":
kept + ch => kept
for i in [0:len(kept) - 1]:
kept[i] => ch
if ch >= "0" and ch <= "9":
out + [int(ch)] => out
elif (ch == "X" or ch == "x") and scheme == 1 and with_check and i == len(kept) - 1:
out + [10] => out
else:
return []
return out
def error_test(scheme, ds):
# Every single wrong digit and every swap of two neighbouring different
# digits, and how many of each the check catches. Returns [singles
# caught, singles, swaps caught, swaps, the swaps that slip through].
0 => s_caught
0 => s_all
len(ds) => n
for i in [0:n - 1]:
9 => top
if scheme == 1 and i == n - 1:
10 => top
for d in [0:top]:
if d != ds[i]:
ds[0:n] => e
d => e[i]
s_all + 1 => s_all
if not valid(scheme, e):
s_caught + 1 => s_caught
0 => t_caught
0 => t_all
[] => missed
for i in [0:n - 2]:
if ds[i] != ds[i + 1]:
ds[0:n] => e
ds[i + 1] => e[i]
ds[i] => e[i + 1]
# an X can only stand last
if not (scheme == 1 and e[n - 2] == 10):
t_all + 1 => t_all
if valid(scheme, e):
missed + [[i, ds[i], ds[i + 1]]] => missed
else:
t_caught + 1 => t_caught
return [s_caught, s_all, t_caught, t_all, missed]
Python projection (checks.py)
def names():
return ["Luhn", "ISBN-10", "ISBN-13 / EAN-13"]
def lengths(scheme):
if scheme == 0:
return [2, 19]
if scheme == 1:
return [10, 10]
return [13, 13]
def luhn_total(ds):
total = 0
n = len(ds)
for i in range(0, n):
d = ds[n - 1 - i]
if i % 2 == 1:
d = d * 2
if d > 9:
d = d - 9
total = total + d
return total
def weighted_total(scheme, ds):
if scheme == 0:
return luhn_total(ds)
total = 0
if scheme == 1:
for i in range(0, 10):
total = total + (10 - i) * ds[i]
return total
for i in range(0, 13):
w = 1
if i % 2 == 1:
w = 3
total = total + w * ds[i]
return total
def modulus(scheme):
if scheme == 1:
return 11
return 10
def valid(scheme, ds):
return weighted_total(scheme, ds) % modulus(scheme) == 0
def check_digit(scheme, body):
top = 10
if scheme == 1:
top = 11
for d in range(0, top):
if valid(scheme, body + [d]):
return d
return -1
def digit_text(d):
if d == 10:
return "X"
return str(d)
def as_text(ds):
s = ""
for d in ds:
s = s + digit_text(d)
return s
def digits_of(scheme, s, with_check):
out = []
kept = ""
for ch in s:
if ch != " " and ch != "-":
kept = kept + ch
for i in range(0, len(kept)):
ch = kept[i]
if ch >= "0" and ch <= "9":
out = out + [int(ch)]
elif (ch == "X" or ch == "x") and scheme == 1 and with_check and i == len(kept) - 1:
out = out + [10]
else:
return []
return out
def error_test(scheme, ds):
s_caught = 0
s_all = 0
n = len(ds)
for i in range(0, n):
top = 9
if scheme == 1 and i == n - 1:
top = 10
for d in range(0, top+1):
if d != ds[i]:
e = ds[0:n]
e[i] = d
s_all = s_all + 1
if not valid(scheme, e):
s_caught = s_caught + 1
t_caught = 0
t_all = 0
missed = []
for i in range(0, n - 2+1):
if ds[i] != ds[i + 1]:
e = ds[0:n]
e[i] = ds[i + 1]
e[i + 1] = ds[i]
if not (scheme == 1 and e[n - 2] == 10):
t_all = t_all + 1
if valid(scheme, e):
missed = missed + [[i, ds[i], ds[i + 1]]]
else:
t_caught = t_caught + 1
return [s_caught, s_all, t_caught, t_all, missed]