Sunday Times Teaser 3314 – Number Unobtainable!
by Stephen Hogg
Published Sunday March 29 2026 (link)
Dyall recalls six-digit phone numbers as a sequence of two three-figure values. His wife, Belle, recalls them as a sequence of three two-figure values. Their courting-days’ phone numbers had no zeroes. For each of their numbers, Dyall recalled that the two three-figure values in order made a descending set of primes, while Belle recalled that the three two-figure values in order made a descending set of primes. Back then their town’s residential numbers never began with “7”, “8” or “9”. Dyall told Tim these things, but not the numbers. He then stated that choosing a digit at random to be told its value would give a one in three chance of Tim being able to work out Dyall’s old phone number with certainty. In Belle’s case the chance was greater still.
Give Belle’s old phone number.
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BRG permalink12345678910111213141516171819202122232425262728293031323334353637from itertools import combinationsfrom collections import Counterprms = {3, 5, 7, 11, 13, 17, 19, 23, 29, 31}prms |= {2} | {n for n in range(37, 1000, 2) if all(n % p for p in prms)}# for collecting phone numbers as digitspnds = []# choose three two digit primes less than 70# in descending order for the phone numbersps = [n for n in range(11, 70, 2) if n in prms][::-1]for p3 in combinations(ps, 3):# collect the six digits of the phone numberdgts = [d for ds in (divmod(n, 10) for n in p3) for d in ds]# convert into two three digit primes ...a, b, c, d, e, f = dgtsr, s = [100 * a + 10 * b + c, 100 * d + 10 * e + f]# ... in descending orderif r > s and {r, s}.issubset(prms):pnds.append(dgts)# collect digits in all phone numbers indexed on their positionspos = list(Counter(dp) for dp in zip(*pnds))# for each digit position in each phone number find the number# of digits that identify the phone numberfor pn in pnds:n = sum(pos[i][d] == 1 for i, d in enumerate(pn))if n == 2:d_pn = ''.join(str(i) for i in pn)elif n > 2:b_pn = ''.join(str(i) for i in pn)else:print(f"Belle: {b_pn} (Dyall: {d_pn}).")
