Sunday Times Teaser 3063 – Pie by Five

by BRG on June 5, 2021

by Colin Vout

Published Sunday June 06 2021 (link)

We had a delicious pie, rectangular and measuring 20 centimetres along the top and 13 centimetres in the other direction. We divided it into five pieces of equal area, using five straight cuts radiating from one internal point. This internal point was rather off centre, in the top left-hand quarter, although the distances from the left and top sides were both a whole number of centimetres. The points where the cuts met the edges were also whole numbers of centimetres along the edges; one edge had two cuts meeting it, and the other three edges had one each.

How far was the internal point from the left and top sides, and how far along the four sides (starting at the top) did the cuts reach the edges (measured clockwise along the edges)?

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Brian Gladman permalink


# the rectangle dimensions and the
# pie slice area in 1/2 cm^2 units
W, H, A = 20, 13, 104

# try rectangular pie in both orientations
for X, Y in ((W, H), (H, W)):  
  # consider the position of the internal
  # point from the left edge ... 
  for x in range(1, (X + 1) // 2):
    # ... and from the top edge
    for y in range(1, (Y + 1) // 2):
      
      # form a triangle slice on the bottom edge
      # with a base of length b 
      b, r = divmod(A, Y - y)
      if r:
        continue
      # position it on the bottom edge
      for a in range(1, X - b):
        
        # form a pie slice on the right edge
        c, r = divmod(A - (X - a - b) * (Y - y), X - x)
        if r or not 0 < c < Y:
          continue
        # form a pie slice on the top edge
        d, r = divmod(A - (Y - c) * (X - x), y)
        if r or not 0 < d < X:
          continue
        # form a pie slice on the left edge
        e, r = divmod(A - (X - d) * y, x)
        if r or not 0 < e < Y:
          continue
        # check that the final slice area is correct
        if (Y - e) * x + a * (Y - y) != A:
          continue
        # the edge distances to the slice lines from the top
        # left corner clockwise
        ep = [X - d, Y - c, (X - a - b, X - a), Y - e]
        print(f"(x, y) = ({x}, {y}); slice edges clockwise {ep}")

# the rectangle dimensions and the

# pie slice area in 1/2 cm^2 units

W, H, A = 20, 13, 104

# try rectangular pie in both orientations

for X, Y in ((W, H), (H, W)):

# consider the position of the internal

# point from the left edge ...

for x in range(1, (X + 1) // 2):

# ... and from the top edge

for y in range(1, (Y + 1) // 2):

# form a triangle slice on the bottom edge

# with a base of length b

b, r = divmod(A, Y - y)

if r:

continue

# position it on the bottom edge

for a in range(1, X - b):

# form a pie slice on the right edge

c, r = divmod(A - (X - a - b) * (Y - y), X - x)

if r or not 0 < c < Y:

continue

# form a pie slice on the top edge

d, r = divmod(A - (Y - c) * (X - x), y)

if r or not 0 < d < X:

continue

# form a pie slice on the left edge

e, r = divmod(A - (X - d) * y, x)

if r or not 0 < e < Y:

continue

# check that the final slice area is correct

if (Y - e) * x + a * (Y - y) != A:

continue

# the edge distances to the slice lines from the top

# left corner clockwise

ep = [X - d, Y - c, (X - a - b, X - a), Y - e]

print(f"(x, y) = ({x}, {y}); slice edges clockwise {ep}")

Frits permalink

@Brian, your program is a bit easier for me to understand than Jim’s.

You can also do the x-loop later as the first check doesn’t deal with x.
It turns out only one value of y is possible.

Reply
Brian Gladman permalink

Yes, I originally had it with the loops in a different order but I decided that it was easier to understand with the x and y loops first.

Reply

Sunday Times Teaser 3063 – Pie by Five

by Colin Vout

Published Sunday June 06 2021 (link)

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