Sunday Times Teaser 2731 – Snarl Up

by Michael Fletcher

Published: 25 January 2015 (link)

I had to drive the six miles from my home into the city centre. The first mile was completed at a steady whole number of miles per hour (and not exceeding the 20mph speed limit). Then each succeeding mile was completed at a lower steady speed than the previous mile, again at a whole number of miles per hour.

After two miles of the journey my average speed had been a whole number of miles per hour, and indeed the same was true after three miles, after four miles, after five miles, and at the end of my journey.

How long did the journey take?

3 Comments Leave one →

Brian Gladman permalink


from fractions import Fraction as RF

# n_leg = the current leg
# cum_t = cumulative time at end of previous leg
# v_max = maximum speed for this leg
# l_mph = speeds for the legs
def solve(n_leg, cum_t, v_max, l_mph):
  # have we completed the six legs?
  if n_leg == 7:
    # return the time for the whole journey (and leg speeds)
    yield cum_t, l_mph    
  else:
    # try the possible speeds for the current leg
    for speed in range(v_max - 1, 6 - n_leg, -1):
      # calculate the total time taken so far
      t = cum_t + RF(1, speed)
      # and the average speed so far
      av = n_leg / t
      # check that this is an integer
      if av.denominator == 1:
        # if so, continue to the next leg
        yield from solve(n_leg + 1, t, speed, l_mph + (speed,))

for t, mph in solve(1, 0, 21, ()):
  print('Journey time was {} hours, leg speeds = {} mph.'.format(t, mph))

from fractions import Fraction as RF

# n_leg = the current leg

# cum_t = cumulative time at end of previous leg

# v_max = maximum speed for this leg

# l_mph = speeds for the legs

def solve(n_leg, cum_t, v_max, l_mph):

# have we completed the six legs?

if n_leg == 7:

# return the time for the whole journey (and leg speeds)

yield cum_t, l_mph

else:

# try the possible speeds for the current leg

for speed in range(v_max - 1, 6 - n_leg, -1):

# calculate the total time taken so far

t = cum_t + RF(1, speed)

# and the average speed so far

av = n_leg / t

# check that this is an integer

if av.denominator == 1:

# if so, continue to the next leg

yield from solve(n_leg + 1, t, speed, l_mph + (speed,))

for t, mph in solve(1, 0, 21, ()):

print('Journey time was {} hours, leg speeds = {} mph.'.format(t, mph))

Don Thompson permalink

Searching to see if anyone had info on this puzzle. I know nothing of programming – trying it longhand and cannot find a solution other than one starting at 24mph – does the programme find one starting below or at 20mph?

Reply
Brian Gladman permalink

Hello Don,

Yes, the solution is as follows:

Journey time was 120 minutes, leg speeds = (20, 12, 6, 5, 2, 1) mph.

Reply

Sunday Times Teaser 2731 – Snarl Up

by Michael Fletcher

Published: 25 January 2015 (link)

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