Sunday Times Teaser 3340 – Generations Jay, Kay, Elle and Em

by BRG on 27 September 2026

by Stephen Hogg

Published Sunday September 27 2026 (link)

Polygonian newborns are allocated a regular polygon talisman for the day of the month they are born (1st = triangle, 2nd = square to 31st = 33-agon). Super-auspicious birth dates occur when the talisman’s internal angle is a whole number of degrees and the month number (Jan = 1 to Dec = 12) divides into it exactly and the year is a multiple of it.

Polygonians, Em, her mam Elle, Elle’s mam Kay and Kay’s dad Jay dined out on Jay’s birthday in 2024. Chef said, “Congratulations, you’re looking fit Jay. You’ll live to 100.” Elle stated they all had super-auspicious birth dates and her 2024 birthday was in the month after next. Em added that she and Kay had their 2024 birthdays in the month before last. Chef then deduced their birth dates with certainty.

Give Em’s birth date

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BRG permalink

Now with changes suggested by Frits to put repetitive code into a function.


from collections import defaultdict

# generational gap
gap = 10

# find all super auspicious days
das = []
for day in range(1, 32):
  angle, r = divmod(180 * day, day + 2)
  if r == 0:
    das.append((day, angle))

# group days by month where the angle divides exactly by the month index
m2vdma = {m: [] for m in range(1, 13)}
for day, angle in das:
  for month in range(1, 13):
    if angle % month == 0:
      m2vdma[month].append((day, month, angle))

# for a person scan their age from <st> to <nd - 1>
# and return valid (day, month, year) values
def find_next(st, nd, mv):
  for y in range(st, nd):
    for d, m, a in m2vdma[mv]:
      if y % a == 0:
        yield d, m, y
  
# store valid combinations of birthdates matching relative
# months indexed on jay's day and month (known by the chef)
sols = defaultdict(list)

# consider J's birthday month (known by chef)
for m in range(3, 11):
  # consider J's birth year
  for jdmy in find_next(1925, 2025 - 3 * gap, m):  
    # consider K's birth year
    for kdmy in find_next(jdmy[2] + gap, 2025 - 2 * gap, m - 2):  
      # consider L's birth year
      for ldmy in find_next(kdmy[2] + gap, 2025 - gap, m + 2):  
        # consider M's birth year
        for mdmy in find_next(ldmy[2] + gap, 2025, m - 2):  
          sols[jdmy[0], jdmy[1]].append([('J', jdmy), ('K', kdmy),
                                         ('L', ldmy), ('M', mdmy)])
# format the output
f = lambda n, dmy: f"{n}:{dmy[0]:02}/{dmy[1]:02}/{dmy[2]}"
# find sets of birth dates that are unique from the chef's perspective
for (jd, jm), ls in sols.items():
  if len(ls) == 1:
    jd, kd, ld, md = ls[0]
    print(f"{f(*jd)}  {f(*kd)}  {f(*ld)}  {f(*md)}")

from collections import defaultdict

# generational gap

gap = 10

# find all super auspicious days

das = []

for day in range(1, 32):

angle, r = divmod(180 * day, day + 2)

if r == 0:

das.append((day, angle))

# group days by month where the angle divides exactly by the month index

m2vdma = {m: [] for m in range(1, 13)}

for day, angle in das:

for month in range(1, 13):

if angle % month == 0:

m2vdma[month].append((day, month, angle))

# for a person scan their age from <st> to <nd - 1>

# and return valid (day, month, year) values

def find_next(st, nd, mv):

for y in range(st, nd):

for d, m, a in m2vdma[mv]:

if y % a == 0:

yield d, m, y

# store valid combinations of birthdates matching relative

# months indexed on jay's day and month (known by the chef)

sols = defaultdict(list)

# consider J's birthday month (known by chef)

for m in range(3, 11):

# consider J's birth year

for jdmy in find_next(1925, 2025 - 3 * gap, m):

# consider K's birth year

for kdmy in find_next(jdmy[2] + gap, 2025 - 2 * gap, m - 2):

# consider L's birth year

for ldmy in find_next(kdmy[2] + gap, 2025 - gap, m + 2):

# consider M's birth year

for mdmy in find_next(ldmy[2] + gap, 2025, m - 2):

sols[jdmy[0], jdmy[1]].append([('J', jdmy), ('K', kdmy),

('L', ldmy), ('M', mdmy)])

# format the output

f = lambda n, dmy: f"{n}:{dmy[0]:02}/{dmy[1]:02}/{dmy[2]}"

# find sets of birth dates that are unique from the chef's perspective

for (jd, jm), ls in sols.items():

if len(ls) == 1:

jd, kd, ld, md = ls[0]

print(f"{f(*jd)} {f(*kd)} {f(*ld)} {f(*md)}")

Frits permalink


from itertools import product, combinations
from collections import defaultdict

clean = lambda d: {k: vs for k, vs in d.items() if vs}

# assume at least 16 years age difference between parent and child
age_diff = 16
max_year = 2020          # Em recalls birthdays

# days in months in 2024
months = (31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31)

# super-auspicious days, months and years 
sa_bdays = {dm[0]: d for d in range(1, 32)
                   if (dm := divmod(d * 180, d + 2))[1] == 0}

sa_months = {a: [m for m in range(1, 13) if a % m == 0] 
                   for a in sa_bdays.keys()} 

sa_years = {a: [y for y in range(1925, max_year + 1) if y % a == 0] 
                  for a, vs in sa_months.items() if vs}

sa_dates = []
for a, years in sa_years.items():
  if not years: continue
  dd = sa_bdays[a]
  # find month/year combinations
  for mm, yyyy in product(sa_months[a], years):
    if dd > months[mm - 1]: 
      continue 
    elif dd == 29 and mm == 2:
      if not (yyyy % 4 == 0 and (yyyy % 100 or yyyy % 400 == 0)):
        continue
    sa_dates.append([mm, dd, yyyy])

# group super-auspicious dates by month
by_month = clean({m : sorted((yyyy, dd) for mm, dd, yyyy in sa_dates 
                      if mm == m) for m in range(1, 13)})   

allmonths = {m for ms in sa_months.values() for m in ms}
# possible months for Jay's birthday
J_months = {m for m in allmonths if m - 2 in allmonths and m + 2 in allmonths}

sols = defaultdict(list)
# check birthdates for Jay
for J_mm, J_dd, J_yyyy in sa_dates:
  if J_mm not in J_months: continue
  if J_yyyy > max_year - 3 * age_diff: continue
  L_mm, K_mm = J_mm + 2, J_mm - 2
  
  # retrieve dates for Kay and Em
  for K, M in combinations(by_month[K_mm], 2):
    K_yyyy, M_yyyy = K[0], M[0]
    if K_yyyy < J_yyyy + age_diff: continue
    if M_yyyy - K_yyyy < 2 * age_diff: continue
    L_range = set(range(K_yyyy + age_diff, M_yyyy - age_diff + 1))
    # try to fit Elle between Kay and Em  
    for L_yyyy, L_dd in by_month[L_mm]:
      if L_yyyy in L_range: 
        sols[(J_dd, J_mm)] += [(J_yyyy, K, (L_yyyy, L_dd), M)]

ems = set()
for k, vs in sols.items():     
  # Chef then deduced their birth dates with certainty
  if len(vs) == 1: 
    y, d = vs[0][-1]
    ems.add(f"{d}-{k[1] - 2}-{y}")

print("answer:", ' or '.join(ems))

from itertools import product, combinations

from collections import defaultdict

clean = lambda d: {k: vs for k, vs in d.items() if vs}

# assume at least 16 years age difference between parent and child

age_diff = 16

max_year = 2020 # Em recalls birthdays

# days in months in 2024

months = (31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31)

# super-auspicious days, months and years

sa_bdays = {dm[0]: d for d in range(1, 32)

if (dm := divmod(d * 180, d + 2))[1] == 0}

sa_months = {a: [m for m in range(1, 13) if a % m == 0]

for a in sa_bdays.keys()}

sa_years = {a: [y for y in range(1925, max_year + 1) if y % a == 0]

for a, vs in sa_months.items() if vs}

sa_dates = []

for a, years in sa_years.items():

if not years: continue

dd = sa_bdays[a]

# find month/year combinations

for mm, yyyy in product(sa_months[a], years):

if dd > months[mm - 1]:

continue

elif dd == 29 and mm == 2:

if not (yyyy % 4 == 0 and (yyyy % 100 or yyyy % 400 == 0)):

continue

sa_dates.append([mm, dd, yyyy])

# group super-auspicious dates by month

by_month = clean({m : sorted((yyyy, dd) for mm, dd, yyyy in sa_dates

if mm == m) for m in range(1, 13)})

allmonths = {m for ms in sa_months.values() for m in ms}

# possible months for Jay's birthday

J_months = {m for m in allmonths if m - 2 in allmonths and m + 2 in allmonths}

sols = defaultdict(list)

# check birthdates for Jay

for J_mm, J_dd, J_yyyy in sa_dates:

if J_mm not in J_months: continue

if J_yyyy > max_year - 3 * age_diff: continue

L_mm, K_mm = J_mm + 2, J_mm - 2

# retrieve dates for Kay and Em

for K, M in combinations(by_month[K_mm], 2):

K_yyyy, M_yyyy = K[0], M[0]

if K_yyyy < J_yyyy + age_diff: continue

if M_yyyy - K_yyyy < 2 * age_diff: continue

L_range = set(range(K_yyyy + age_diff, M_yyyy - age_diff + 1))

# try to fit Elle between Kay and Em

for L_yyyy, L_dd in by_month[L_mm]:

if L_yyyy in L_range:

sols[(J_dd, J_mm)] += [(J_yyyy, K, (L_yyyy, L_dd), M)]

ems = set()

for k, vs in sols.items():

# Chef then deduced their birth dates with certainty

if len(vs) == 1:

y, d = vs[0][-1]

ems.add(f"{d}-{k[1] - 2}-{y}")

print("answer:", ' or '.join(ems))

Frits permalink


from collections import defaultdict
from itertools import combinations

# generational gap
gap = 10

# super auspicious days per angle
angles = {dm[0]: d for d in range(1, 32)
                 if (dm := divmod(d * 180, d + 2))[1] == 0}

# feasible years 
years = [(y, a, angles[a]) for y in range(1925, 2021) for a in angles.keys()
         if y % a == 0]  

sols = defaultdict(list)      

# select years for Jay, Kay, Elle and Em          
for c in combinations(years, 4):
  if any(y[0] - x[0] < gap for x, y in zip(c, c[1:])): continue
  # feasible months for Jay, Kay, Elle and Em
  mss = [{m for m in range(1, 13) if a % m == 0} for _, a, _ in c]
  # process months for Jay
  for jm in mss[0]:
    if (jm - 2) not in mss[1] or (jm - 2) not in mss[3]: continue # Kay and Em
    if jm + 2 not in mss[2]: continue # Elle
    # days for Jay, Kay, Elle and Em
    days = [x[2] for x in c]
    ym, jd = c[3][0], c[0][2]
    sols[(jm, jd)].append((ym, days))
 
ems = set()
for k, vs in sols.items():     
  # Chef then deduced their birth dates with certainty
  if len(vs) == 1: 
    y, d = vs[0][0], vs[0][-1][-1]
    ems.add(f"{d}-{k[0] - 2}-{y}")
 
print("answer:", ' or '.join(ems))

from collections import defaultdict

from itertools import combinations

# generational gap

gap = 10

# super auspicious days per angle

angles = {dm[0]: d for d in range(1, 32)

if (dm := divmod(d * 180, d + 2))[1] == 0}

# feasible years

years = [(y, a, angles[a]) for y in range(1925, 2021) for a in angles.keys()

if y % a == 0]

sols = defaultdict(list)

# select years for Jay, Kay, Elle and Em

for c in combinations(years, 4):

if any(y[0] - x[0] < gap for x, y in zip(c, c[1:])): continue

# feasible months for Jay, Kay, Elle and Em

mss = [{m for m in range(1, 13) if a % m == 0} for _, a, _ in c]

# process months for Jay

for jm in mss[0]:

if (jm - 2) not in mss[1] or (jm - 2) not in mss[3]: continue # Kay and Em

if jm + 2 not in mss[2]: continue # Elle

# days for Jay, Kay, Elle and Em

days = [x[2] for x in c]

ym, jd = c[3][0], c[0][2]

sols[(jm, jd)].append((ym, days))

ems = set()

for k, vs in sols.items():

# Chef then deduced their birth dates with certainty

if len(vs) == 1:

y, d = vs[0][0], vs[0][-1][-1]

ems.add(f"{d}-{k[0] - 2}-{y}")

print("answer:", ' or '.join(ems))

Sunday Times Teaser 3340 – Generations Jay, Kay, Elle and Em

by Stephen Hogg

Published Sunday September 27 2026 (link)

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