Sunday Times Teaser 2879 – Blood Line

by Danny Roth

Published November 26 2017 (link)

Patients numbered 1 to 99 were waiting for a blood test. George’s and Martha’s numbers were consecutive. Patients were called in random order to one of the available cubicles numbered 1 to 9. The nurse in each cubicle took a fixed whole number of minutes to do the test, but that time varied from cubicle to cubicle. All cubicles started at 10am and they all finished together before 5pm on the same day. Amazingly, it turned out that each patient’s cubicle number was the highest digit that divided into their patient number. George noted that, when he was called, the session had been running for a number of minutes whose digits added to his cubicle number. Martha noted that the same was true for her.

What were their patient numbers?

One Comment Leave one →

Brian Gladman permalink

This teaser appears to have a flaw as there are five possible solutions.


from math import gcd

# create lists of patients for each cubicle
c2p = [[] for cubicle in range(1, 11)]
for patient in range(1, 100):
  for cubicle in range(9, 0, -1):
    if not patient % cubicle:
      c2p[cubicle].append(patient)
      break

# find the overall time for all patients, noting that this
# is the same for all cubicles;  hence the overall time is
# the product of the number of patients for a cubicule and
# the cubicle's time per patient; so the overall time must
# be the lowest common multiple of the numbers of patients
# per cubicle (or a multiple of this)
time = 1
for cubicle in range(1, 10):
  ppc = len(c2p[cubicle])
  time *= ppc // gcd(time, ppc)

total_time = 0
while True:
  total_time += time
  # the total time is less than seven hours
  if total_time >= 7 * 60:
    break

  # for c in range(1, 10):
  #   tm = total_time // len(c2p[c])
  #   print(f'{c} {len(c2p[c]):2} {tm:2} {c2p[c]}')
  
  # find patient numbers for which the digit sum of 
  # their treatment time equals their cubicle number
  nbrs = set()
  for cubicle in range(1, 10):
    # the time for treatment in this cubicle
    ct = total_time // len(c2p[cubicle])
    # consider all possible treatment start times
    for t in range(0, total_time, ct):
      if sum(divmod(t // 10, 10)) + t % 10 == cubicle:
        nbrs.update(c2p[cubicle])
  # now find consecutive patient numbers
  print([(x, x + 1) for x in sorted(nbrs) if x + 1 in nbrs])

from math import gcd

# create lists of patients for each cubicle

c2p = [[] for cubicle in range(1, 11)]

for patient in range(1, 100):

for cubicle in range(9, 0, -1):

if not patient % cubicle:

c2p[cubicle].append(patient)

break

# find the overall time for all patients, noting that this

# is the same for all cubicles; hence the overall time is

# the product of the number of patients for a cubicule and

# the cubicle's time per patient; so the overall time must

# be the lowest common multiple of the numbers of patients

# per cubicle (or a multiple of this)

time = 1

for cubicle in range(1, 10):

ppc = len(c2p[cubicle])

time *= ppc // gcd(time, ppc)

total_time = 0

while True:

total_time += time

# the total time is less than seven hours

if total_time >= 7 * 60:

break

# for c in range(1, 10):

# tm = total_time // len(c2p[c])

# print(f'{c} {len(c2p[c]):2} {tm:2} {c2p[c]}')

# find patient numbers for which the digit sum of

# their treatment time equals their cubicle number

nbrs = set()

for cubicle in range(1, 10):

# the time for treatment in this cubicle

ct = total_time // len(c2p[cubicle])

# consider all possible treatment start times

for t in range(0, total_time, ct):

if sum(divmod(t // 10, 10)) + t % 10 == cubicle:

nbrs.update(c2p[cubicle])

# now find consecutive patient numbers

print([(x, x + 1) for x in sorted(nbrs) if x + 1 in nbrs])

Sunday Times Teaser 2879 – Blood Line

by Danny Roth

Published November 26 2017 (link)

Leave a Reply Cancel reply

Recent Posts

Recent Comments

Archives

Categories

Meta