Sunday Times Teaser 3010 – Putting Game

by John Owen

Published Sunday May 31 2020 (link)

A putting game has a board with eight rectangular holes, like the example (not to scale), but with the holes in a different order.

If you hit your ball (diameter 4cm) through a hole without touching the sides, you score the number of points above that hole. The sum of score and width (in cm) for each hole is 15; there are 2cm gaps between holes.

I know that if I aim at a point on the board, then the ball’s centre will arrive at the board within 12cm of my point of aim, and is equally likely to arrive at any point in that range. Therefore, I aim at the one point that maximises my long-term average score. This point is the centre of a hole and my average score is a whole number.

(a) Which hole do I aim at?

(b) Which two holes are either side of it?

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Brian Gladman permalink


from itertools import combinations

# find the overlap between two ranges of x values
def overlap(b, h):
  al, ar = b
  bl, br = h
  return max(0, min(ar, br) - max(al, bl))

# the centre hole score (sc) and its length (hc)  
for sc in range(1, 9):
  hc = 15 - sc
  # the adjacent holes and their lengths
  for sl, sr in combinations(sorted(set(range(1, 9)).difference([sc])), 2):
    hl, hr = 15 - sl, 15 - sr

    # list the score and the ranges (start, end) of the holes, measured from the
    # centre of the central hole in quarter centimetre units
    t, u = 2 * hc - 8, 2 * hc + 16
    holes = [(-(2 * hc + 4 * hl), -u, sl), (-t, t, sc), (u, 2 * hc + 4 * hr, sr)]
    
    sol = []
    # vary the point aimed at
    for ap in range(-20, 21):
      # ... and compute the score for each point
      br = (ap - 48, ap + 48)
      tot = sum(v * overlap(br, (l, r)) for l, r, v in holes)
      sol.append((tot, ap))
 
    # there must be only one maximum score
    smx = max(s for s, p in sol)
    sol = list(s for s in sol if s[0] == smx)
    if len(sol) == 1:
      tot, ap = sol[0]
      # find the average score
      av, r = divmod(tot, 96)
      # the score is an integer, achieved by aiming at the centre of a hole
      if not r and ap in  {(x + y) // 2 for x, y, v in holes}:
        print(f"(a) {sc}; (b) ({sl}, {sr}) giving an average score of {av}")

from itertools import combinations

# find the overlap between two ranges of x values

def overlap(b, h):

al, ar = b

bl, br = h

return max(0, min(ar, br) - max(al, bl))

# the centre hole score (sc) and its length (hc)

for sc in range(1, 9):

hc = 15 - sc

# the adjacent holes and their lengths

for sl, sr in combinations(sorted(set(range(1, 9)).difference([sc])), 2):

hl, hr = 15 - sl, 15 - sr

# list the score and the ranges (start, end) of the holes, measured from the

# centre of the central hole in quarter centimetre units

t, u = 2 * hc - 8, 2 * hc + 16

holes = [(-(2 * hc + 4 * hl), -u, sl), (-t, t, sc), (u, 2 * hc + 4 * hr, sr)]

sol = []

# vary the point aimed at

for ap in range(-20, 21):

# ... and compute the score for each point

br = (ap - 48, ap + 48)

tot = sum(v * overlap(br, (l, r)) for l, r, v in holes)

sol.append((tot, ap))

# there must be only one maximum score

smx = max(s for s, p in sol)

sol = list(s for s in sol if s[0] == smx)

if len(sol) == 1:

tot, ap = sol[0]

# find the average score

av, r = divmod(tot, 96)

# the score is an integer, achieved by aiming at the centre of a hole

if not r and ap in {(x + y) // 2 for x, y, v in holes}:

print(f"(a) {sc}; (b) ({sl}, {sr}) giving an average score of {av}")

Erling Torkildsen has produced a nice model for playing this game here

Sunday Times Teaser 3010 – Putting Game

by John Owen

Published Sunday May 31 2020 (link)

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