Sunday Times Teaser 3343 – Cartesian Darts
by Mark Valentine
Published Sunday October 18 2026 (link)
An AI app is learning to play a game of virtual darts. The board is a square, subdivided into nine squares. The resulting central square is divided in the same way, and the process continues ad infinitum.
The value of each outer square is 1. Each smaller square has the same constant prime multiple of the value of the next larger square. A dart landing in a given square scores the average value of all squares of the same size or smaller, weighted by area. Thus, the closer the dart lands to the centre, the higher the score.
At first, the app sometimes got a non-integer score, but after improving always got an integer score.
What was the non-integer score (as a fraction)?
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BRG permalink123456789101112131415161718192021222324252627282930313233# The outer square dart score can be derived from:## S[0] = (8 / 9) + p.S[0] / 9## which gives:## S[0] = 8 / (9 - p)## For each level the score inceases by a factor of p giving:## S[n] = (8 / (9 - p)).p^nfrom fractions import Fraction# consider primes less than 9for p in (2, 3, 5, 7):score = Fraction(8, 9 - p)# compute up to five scores looking for a fractional# score followed by an integer onefor n in range(5):# compute the next score and exit the loop if an integerif (next := p * score).denominator == 1:# give the answer if the previious score is a fractionif score.denominator > 1:print('Answer:', score)breakscore = nextelse:# go to next primecontinue
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GeoffR permalink1234567891011121314151617from fractions import Fraction# Outer ring is 8/9 of the board, worth 1. The centre (1/9) is a copy of# the board worth p times as much, so the board average A = 8/9 + p*A/9,# giving A = 8 / (9 - p). Each level in multiplies the score by p, so# level n scores 8 * p**n / (9 - p). Needs p < 9 for A to be positive.# If s*p is an integer, every later score s*p**k = (s*p) * p**(k-1) is# an integer times an integer, so "always an integer" needs only s*p.for p in [2, 3, 5, 7]:s = Fraction(8, 9 - p)if s.denominator != 1 and (s * p).denominator == 1:print("Prime multiple p =", p)print("Non-integer score (outer ring) =", s)print("Next score in (integer) =", s * p)
