Sunday Times Teaser 3147 – Noteworthy
by Victor Bryant
Published Sunday January 15 2023 (link)
Apparently in Costa Lotta a single-digit percentage of banknotes are forgeries and so I have designed a marker pen which tests whether notes are genuine. I thought it would be quite useful to the banks because, on average, for every N uses it only gives an incorrect result once (where N is some whole number).
Unfortunately my design has been abandoned by the banks because it turns out that on average for every N occasions on which the pen indicates a forgery, only one of the notes will in fact be forged!
What is N?
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BRG permalink12345678910111213141516171819202122232425262728from math import isqrt# b = fraction of notes that are forgeries (bad); f = fraction# of notes wrongly classified; t = number of tests run## 1. genuine notes wrongly tested as forgeries = f.(1 - b).t# 2. genuine notes correctly tested as genuine = (1 - f)(1 - b).t# 3. forged notes wrongly tested as genuine = f.b.t# 4. forged notes correctly tested as forged = (1 - f).b.t## Test error rate (1 + 3): b.t = t / N ==> f = 1 / N## Of tests indicating forgeries (1 + 4) only 1 was correct:# f.(1- b).t = N - 1 ; (1 - f).b.t = 1# ==> f.(1 - b) = (N - 1)(1 - f).b## Eliminating f and simpifying: 1 - b = (N - 1)^2.b## ==> N = sqrt((1 - b) / b) + 1# ==> N = sqrt(100 / d - 1) + 1 (with b = d / 100)#for d in range(1, 10):if 100 % d == 0:t = 100 // d - 1if (r := isqrt(t)) ** 2 == t:print(f"N = {r + 1} ({d = })")
