Sunday Times Teaser 3231 – In His Prime

by BRG on 25 August 2024

by Victor Bryant

Published Sunday August 25 2024 (link)

Once, on my grandson’s birthday, I asked him the following five questions:

How many days are there in this month?

How many Mondays are there in this month?

How many “prime” days (ie, 2nd, 3rd, 5th, …) are there in this month?

How many of those prime days are Saturdays?

How many letters are there when you spell the month?

The total of the five answers was a prime number.

Then I asked him the same questions the next day. All five answers were different [1] and no answer was the same as for the day before, but again the total of the five answers was prime.

When I first asked the questions what was the month and day of the week?

[1] Not in the teaser as originally published.

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BRG permalink


import calendar as cal

pr = {3, 5, 7}
prms = {2} | pr | {n for n in range(11, 100, 2) if all(n % p for p in pr)}

# return answers to the questions for a given date
def answers(month, year):
  q = [0] * 5
  week_day, month_length = cal.monthrange(year, month)
  # the month length and month name length
  q[0] = month_length
  q[4] = len(cal.month_name[month])
  # count Mondays, prime days and prime Saturdays
  for n in range(1, month_length + 1):
    q[1] += (week_day == cal.MONDAY)
    q[2] += (n in prms)
    q[3] += (n in prms and week_day == cal.SATURDAY)
    week_day = (week_day + 1) % 7
  return tuple(q)

previous_month = lambda m, y: (m - 1, y) if m > 1 else (12, y - 1)

def solutions(min_year):
  # go back in time one month at a time 
  month, year = 8, 2024
  while year >= min_year:
    month, year = previous_month(month, year)
    
    # find the answers and check that they have a prime sum
    if sum((nd_ans := answers(month, year))) in prms:
    
      # the birthday date
      pm, py = previous_month(month, year)
      
      # check that the birthday answers have a prime sum 
      if sum((bd_ans := answers(pm, py))) in prms:
    
        # check that no answers are the same on the two days
        if not any(a == b for a, b in zip(bd_ans, nd_ans)):
          yield cal.monthrange(py, pm)[1], pm, py, bd_ans, nd_ans

for bd, bm, by, bd_ans, nd_ans in solutions(2000):
  dn, mn = cal.day_name[cal.weekday(by, bm, bd)], cal.month_name[bm]
  print(f"{dn} {mn} {bd} {by} [{bd_ans}, {nd_ans}].")

import calendar as cal

pr = {3, 5, 7}

prms = {2} | pr | {n for n in range(11, 100, 2) if all(n % p for p in pr)}

# return answers to the questions for a given date

def answers(month, year):

q = [0] * 5

week_day, month_length = cal.monthrange(year, month)

# the month length and month name length

q[0] = month_length

q[4] = len(cal.month_name[month])

# count Mondays, prime days and prime Saturdays

for n in range(1, month_length + 1):

q[1] += (week_day == cal.MONDAY)

q[2] += (n in prms)

q[3] += (n in prms and week_day == cal.SATURDAY)

week_day = (week_day + 1) % 7

return tuple(q)

previous_month = lambda m, y: (m - 1, y) if m > 1 else (12, y - 1)

def solutions(min_year):

# go back in time one month at a time

month, year = 8, 2024

while year >= min_year:

month, year = previous_month(month, year)

# find the answers and check that they have a prime sum

if sum((nd_ans := answers(month, year))) in prms:

# the birthday date

pm, py = previous_month(month, year)

# check that the birthday answers have a prime sum

if sum((bd_ans := answers(pm, py))) in prms:

# check that no answers are the same on the two days

if not any(a == b for a, b in zip(bd_ans, nd_ans)):

yield cal.monthrange(py, pm)[1], pm, py, bd_ans, nd_ans

for bd, bm, by, bd_ans, nd_ans in solutions(2000):

dn, mn = cal.day_name[cal.weekday(by, bm, bd)], cal.month_name[bm]

print(f"{dn} {mn} {bd} {by} [{bd_ans}, {nd_ans}].")

Frits permalink

Allowing for multiple solutions.
The “next day” must be in a different month.


from datetime import date, timedelta

# primes below 70 (should be enough)
P = {3, 5, 7}
P |= {2} | {n for n in range(11, 70, 2) if all(n % p for p in P)}

prevmonth = lambda y, m: (y, m - 1) if m > 1 else (y - 1, 12)

# return all days in date format for a specific month
def gen_dates(y, m):
  d = date(y, m, 1)
  inc = timedelta(days=1)
  s = []
  while d.month == m:
    s.append(d)
    d += inc
  return s  

# get answers to the questions (input: dates in the month)
def get_answers(ds):  
  # 1. number of days in the month
  a1 = len(ds)
  # 2. number of Mondays in this month
  a2 = len([1 for d in ds if d.isoweekday() == 1])
  # 3. number of prime days in this month
  a3 = len([1 for d in ds if d.day in P])
  # 4. number of prime days on a Saturday in this month
  a4 = len([1 for d in ds if d.day in P and d.isoweekday() == 6])
  # 5. number of letters in this month
  a5 = len(ds[0].strftime('%B'))
  return (a1, a2, a3, a4, a5)

(y, m, sols) = (2024, 7, set())
(prev_n, prev_a, prev_p5) = (12 * y + m + 1, [], False)
# loop backwards in time
while y > 1912:  # allowing for old grandchildren (min/max mothers 5/116)
  # get dates for month <m> in year <y>
  dates = gen_dates(y, m)
  # total of the five answers was a prime number
  if (prime5 := sum(answs := get_answers(dates)) in P): 
    # two consecutive months where total of five answers was prime
    if prev_p5 and 12 * y + m == prev_n - 1:
      # no answer was the same
      if all(x != y for x, y in zip(answs, prev_a)): 
        sols.add((dates[-1].strftime('%B'), dates[-1].strftime('%A')))
    
  # store results of this month 
  (prev_n, prev_a, prev_p5) = (prev_n - 1, answs, prime5)
  (y, m) = prevmonth(y, m)

print("answer(s):")
print('   or\n'.join(f"  month = {m}, day of the week = {d}" for m, d in sols))

from datetime import date, timedelta

# primes below 70 (should be enough)

P = {3, 5, 7}

P |= {2} | {n for n in range(11, 70, 2) if all(n % p for p in P)}

prevmonth = lambda y, m: (y, m - 1) if m > 1 else (y - 1, 12)

# return all days in date format for a specific month

def gen_dates(y, m):

d = date(y, m, 1)

inc = timedelta(days=1)

s = []

while d.month == m:

s.append(d)

d += inc

return s

# get answers to the questions (input: dates in the month)

def get_answers(ds):

# 1. number of days in the month

a1 = len(ds)

# 2. number of Mondays in this month

a2 = len([1 for d in ds if d.isoweekday() == 1])

# 3. number of prime days in this month

a3 = len([1 for d in ds if d.day in P])

# 4. number of prime days on a Saturday in this month

a4 = len([1 for d in ds if d.day in P and d.isoweekday() == 6])

# 5. number of letters in this month

a5 = len(ds[0].strftime('%B'))

return (a1, a2, a3, a4, a5)

(y, m, sols) = (2024, 7, set())

(prev_n, prev_a, prev_p5) = (12 * y + m + 1, [], False)

# loop backwards in time

while y > 1912: # allowing for old grandchildren (min/max mothers 5/116)

# get dates for month <m> in year <y>

dates = gen_dates(y, m)

# total of the five answers was a prime number

if (prime5 := sum(answs := get_answers(dates)) in P):

# two consecutive months where total of five answers was prime

if prev_p5 and 12 * y + m == prev_n - 1:

# no answer was the same

if all(x != y for x, y in zip(answs, prev_a)):

sols.add((dates[-1].strftime('%B'), dates[-1].strftime('%A')))

# store results of this month

(prev_n, prev_a, prev_p5) = (prev_n - 1, answs, prime5)

(y, m) = prevmonth(y, m)

print("answer(s):")

print(' or\n'.join(f" month = {m}, day of the week = {d}" for m, d in sols))

Sunday Times Teaser 3231 – In His Prime

by Victor Bryant

Published Sunday August 25 2024 (link)

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