Sunday Times Teaser 2600 – Mileometer

by John Owen

Published: 22 July 2012 (link)

My car’s digital mileometer has five digits, displaying whole miles. It also has a trip meter with five digits, displaying tenths of a mile (which resets to zero when reaching 10,000 miles). Soon after the car was new I reset the trip meter to zero (but never did that again). Not long after, I noticed that the two five-digit displays were the same (ignoring the decimal point). The next time the displays were the same I wrote down the mileage and did so on each subsequent occasion until the mileage reached 100,000. The sum of all the mileages I wrote down was 500,000.

What were the five digits when the displays were first the same?

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brian gladman permalink

Here is my solution:


# Let the  miles be m miles and n tenths of a mile (m.n) and
# let the  mileage  when the  trip meter is reset be r miles
# and s tenths of a mile. After the reset the the trip meter
# reading t (u miles and v tenths of a mile) is given by:
#
#   t = 10.u + v = 10.m + n - 10.r - s 
#   
# The  mileometer and the trip meter are the same whenever m 
# is equal to 10.u + v mod 100000: 
#
#   9.m = 10.r + s - n mod 100000
#
# Multiplying this by 11111 and taking the result mod 100000
# gives:
#
#   m = 88890.r + 11111.(n - s) mod 100000
#
# As n goes from 0..9, there are  ten values  where  the two
# meters read the same.   All but the smallest of these must
# sum to 500000

# output format
fs = 'First match = {:05d} (trip meter reset at {:0d}.{:d}).'

# test for a reset in the first 5000 miles
for reset_point in range(50000):
  # trip meter miles (r) and tenths (s)
  r, s = divmod(reset_point, 10)
  # list the ten values of m where the meters read the same
  l = [(88890 * r + 11111 * (n - s)) % 100000 for n in range(10)]
  # check that the sum of all but the smallest is 500,000
  if sum(l) - min(l) == 500000:
    print(fs.format(min(l), r, s))

# Let the miles be m miles and n tenths of a mile (m.n) and

# let the mileage when the trip meter is reset be r miles

# and s tenths of a mile. After the reset the the trip meter

# reading t (u miles and v tenths of a mile) is given by:

# t = 10.u + v = 10.m + n - 10.r - s

# The mileometer and the trip meter are the same whenever m

# is equal to 10.u + v mod 100000:

# 9.m = 10.r + s - n mod 100000

# Multiplying this by 11111 and taking the result mod 100000

# gives:

# m = 88890.r + 11111.(n - s) mod 100000

# As n goes from 0..9, there are ten values where the two

# meters read the same. All but the smallest of these must

# sum to 500000

# output format

fs = 'First match = {:05d} (trip meter reset at {:0d}.{:d}).'

# test for a reset in the first 5000 miles

for reset_point in range(50000):

# trip meter miles (r) and tenths (s)

r, s = divmod(reset_point, 10)

# list the ten values of m where the meters read the same

l = [(88890 * r + 11111 * (n - s)) % 100000 for n in range(10)]

# check that the sum of all but the smallest is 500,000

if sum(l) - min(l) == 500000:

print(fs.format(min(l), r, s))

Sunday Times Teaser 2600 – Mileometer

by John Owen

Published: 22 July 2012 (link)

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