Sunday Times Teaser 3319 – Watch Out!

by BRG on May 3, 2026

by Victor Bryant

Published Sunday May 03 2026 (link)

One January a few years ago I was given a new watch that also displayed the date. It did this by turning two wheels, one with 0-3 on it and the other with 0-9. So, if left running, it displayed in turn 00, 01, 02, … 29, 30, 31, … 39, 00, 01, … . Therefore it needed resetting each month, but I never bothered. The date was correctly displayed when I received the watch but then was incorrect for every other month of that year. However, on my birthday that year the display showed an odd number that was in fact the correct date but with the digits reversed! I had to console myself with the fact that, before the watch was 30 years old, on one occasion my birthday would be correctly displayed!

When is my birthday (day and month)?

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BRG permalink

This is based on the approach followed by Jim Randell here.

from datetime import date, timedelta

# convert a date to its watch display
day_zero = 0
watch = lambda d: (d.toordinal() - day_zero) % 40

# consider the year the watch was gifted (avoid the 2024 leap year!)
for w_year in range(2018, 2024):
  poss_bdays = set()
  # set day 1 and day zero reference points
  cd = date(day = 1, month = 1, year = w_year)
  day_zero = cd.toordinal() - 1
  # look for possible birthdays in the first year
  while cd.year == w_year:
    # the watch display must be wrong for months after January    
    if (w_dsp := watch(cd)) == cd.day and cd.month > 1:
      poss_bdays = set()
      break
    # reverse the current two digit day to give a watch display ...
    rev_val = int(f"{cd.day:02}"[::-1])
    # ... and look for an odd watch display that matches this
    if w_dsp % 2 == 1 and w_dsp == rev_val:
      poss_bdays.add((cd, w_dsp))
    # move to the next day
    cd += timedelta(days = 1)

# now consider birthdays in the next 29 years
  for bd, w_dsp in poss_bdays:
    while bd.year < w_year + 30:
      bd = bd.replace(year = bd.year + 1)
      if (wd := watch(bd)) == bd.day:
        print(f"{bd.strftime('%dth %B')} "
              f"[({cd.year - 1} watch {w_dsp:02d}) -> ({bd.year} watch {wd:02d})]")

from datetime import date, timedelta

# convert a date to its watch display

day_zero = 0

watch = lambda d: (d.toordinal() - day_zero) % 40

# consider the year the watch was gifted (avoid the 2024 leap year!)

for w_year in range(2018, 2024):

poss_bdays = set()

# set day 1 and day zero reference points

cd = date(day = 1, month = 1, year = w_year)

day_zero = cd.toordinal() - 1

# look for possible birthdays in the first year

while cd.year == w_year:

# the watch display must be wrong for months after January

if (w_dsp := watch(cd)) == cd.day and cd.month > 1:

poss_bdays = set()

break

# reverse the current two digit day to give a watch display ...

rev_val = int(f"{cd.day:02}"[::-1])

# ... and look for an odd watch display that matches this

if w_dsp % 2 == 1 and w_dsp == rev_val:

poss_bdays.add((cd, w_dsp))

# move to the next day

cd += timedelta(days = 1)

# now consider birthdays in the next 29 years

for bd, w_dsp in poss_bdays:

while bd.year < w_year + 30:

bd = bd.replace(year = bd.year + 1)

if (wd := watch(bd)) == bd.day:

print(f"{bd.strftime('%dth %B')} "

f"[({cd.year - 1} watch {w_dsp:02d}) -> ({bd.year} watch {wd:02d})]")

Frits permalink

Not very efficient (no logic to limit years nor performing the 30 years logic for specific birthdays).

from datetime import datetime, timedelta

# return birthdays for days up to year <end_year> with special logic
def get_bdays(start_year, end_year, excl_years=set()):
  dt = datetime(start_year, 1, 1)
  # on my birthday that year the display showed an odd number that was 
  # in fact the correct date but with the digits reversed
  w = 1 # set start of the watch
  bdays, dt_ = [], dt
  while dt_.year <= end_year:
    dt_ += timedelta(days=1)
    w = (w + 1) % 40
    
    if dt_.year in excl_years: continue
    
    if excl_years: # exact match with watch
      d_, w_ = dt_.day, w
    else:
      # w is an odd number and only correct when reversed
      if (w % 10) not in {1, 3} or w == 11:
        continue
    
      # reverse digits
      d_, w_ = str(dt_.day).zfill(2)[::-1], str(w).zfill(2)

# store birthdays 
    if d_ == w_:
      bdays.append(100 * dt_.month + dt_.day) 
  return bdays

sols = set()
# a few years ago
for yyyy in range(2010, 2024):
  d0 = datetime(yyyy, 1, 1)
  d = d0
  w = 1 # set start of the watch

# check days for the other months of the year
  while d.year == d0.year:
    d += timedelta(days=1)
    w = (w + 1) % 40
    if d.day == w and d.month > d0.month:
      break

# if incorrect for every other month of that year
  if d.year > d0.year:
    # reverse digits logic for year yyyy
    bdays = get_bdays(yyyy, yyyy)
    if not bdays: continue
    
    # exact match with watch for the the following 29 years
    seen30 = get_bdays(yyyy, yyyy + 29, {yyyy})
    # store birthdays seen once in the following 29 years
    sols.update(set(str(bd) for bd in bdays if seen30.count(bd) == 1))

print(f"answer: {' or '.join(s[-2:] + '-' + s[:-2] for s in sols)}")

from datetime import datetime, timedelta

# return birthdays for days up to year <end_year> with special logic

def get_bdays(start_year, end_year, excl_years=set()):

dt = datetime(start_year, 1, 1)

# on my birthday that year the display showed an odd number that was

# in fact the correct date but with the digits reversed

w = 1 # set start of the watch

bdays, dt_ = [], dt

while dt_.year <= end_year:

dt_ += timedelta(days=1)

w = (w + 1) % 40

if dt_.year in excl_years: continue

if excl_years: # exact match with watch

d_, w_ = dt_.day, w

else:

# w is an odd number and only correct when reversed

if (w % 10) not in {1, 3} or w == 11:

continue

# reverse digits

d_, w_ = str(dt_.day).zfill(2)[::-1], str(w).zfill(2)

# store birthdays

if d_ == w_:

bdays.append(100 * dt_.month + dt_.day)

return bdays

sols = set()

# a few years ago

for yyyy in range(2010, 2024):

d0 = datetime(yyyy, 1, 1)

d = d0

w = 1 # set start of the watch

# check days for the other months of the year

while d.year == d0.year:

d += timedelta(days=1)

w = (w + 1) % 40

if d.day == w and d.month > d0.month:

break

# if incorrect for every other month of that year

if d.year > d0.year:

# reverse digits logic for year yyyy

bdays = get_bdays(yyyy, yyyy)

if not bdays: continue

# exact match with watch for the the following 29 years

seen30 = get_bdays(yyyy, yyyy + 29, {yyyy})

# store birthdays seen once in the following 29 years

sols.update(set(str(bd) for bd in bdays if seen30.count(bd) == 1))

print(f"answer: {' or '.join(s[-2:] + '-' + s[:-2] for s in sols)}")

John Z permalink

mo = ((31, 31), (28, 29), (31, 31), (30, 30), (31, 31), (30, 30),
      (31, 31), (31, 31), (30, 30), (31, 31), (30, 30), (31, 31))

# consider initial year "a few years ago"
for iy in (2022, 2023, 2024, 2025):
  matchflag = 0
  # initialise possible birthdays when wd is inverse of day date
  pbd = []
  # candidate birthdays
  cbd = [[] for i in range(4)]

# initialise watch display
  wd = 0
  # consider up to thirty years more
  for yc in range(30):
    #leap year (1) or not (0)
    ly = int((yc + iy) % 4 == 0)

#consider months
    for m in range(12):
      # consider days taking into account leap years
      for d in range(1, mo[m][ly] + 1):
        # increase watch display by 1
        wd = (wd + 1) % 40
        # if initial year check for potential birthdays
        if yc == 0:
          # day and watch display must not correspond in initial year after Jan
          if m and wd == d:
            matchflag = 1 # (finish out year before checking flag again)

# must not be January, display is odd and inverse is equal
          if (m  and wd & 1 and str(wd).zfill(2) == str(d).zfill(2)[::-1]):
            # save day and month
            pbd += [(d , m + 1)]
        # check for correct watch display in subsequent years 
        else:
          if wd == d and any(d == a and m + 1 == b for a, b in pbd):
          # save in candidate birthdays
            cbd[iy - 2022] += [(iy, yc, d, m + 1)]
    if matchflag:
      break
  # no watch display and date matches were found in this year after Jan
  else:
    for cc in cbd:
      if len(cc) == 1:
        print('Initial year:', cc[0][0], 'number of years:', cc[0][1],
              'birthday:', cc[0][2], cc[0][3])

mo = ((31, 31), (28, 29), (31, 31), (30, 30), (31, 31), (30, 30),

(31, 31), (31, 31), (30, 30), (31, 31), (30, 30), (31, 31))

# consider initial year "a few years ago"

for iy in (2022, 2023, 2024, 2025):

matchflag = 0

# initialise possible birthdays when wd is inverse of day date

pbd = []

# candidate birthdays

cbd = [[] for i in range(4)]

# initialise watch display

wd = 0

# consider up to thirty years more

for yc in range(30):

#leap year (1) or not (0)

ly = int((yc + iy) % 4 == 0)

#consider months

for m in range(12):

# consider days taking into account leap years

for d in range(1, mo[m][ly] + 1):

# increase watch display by 1

wd = (wd + 1) % 40

# if initial year check for potential birthdays

if yc == 0:

# day and watch display must not correspond in initial year after Jan

if m and wd == d:

matchflag = 1 # (finish out year before checking flag again)

# must not be January, display is odd and inverse is equal

if (m and wd & 1 and str(wd).zfill(2) == str(d).zfill(2)[::-1]):

# save day and month

pbd += [(d , m + 1)]

# check for correct watch display in subsequent years

else:

if wd == d and any(d == a and m + 1 == b for a, b in pbd):

# save in candidate birthdays

cbd[iy - 2022] += [(iy, yc, d, m + 1)]

if matchflag:

break

# no watch display and date matches were found in this year after Jan

else:

for cc in cbd:

if len(cc) == 1:

print('Initial year:', cc[0][0], 'number of years:', cc[0][1],

'birthday:', cc[0][2], cc[0][3])

Sunday Times Teaser 3319 – Watch Out!

by Victor Bryant

Published Sunday May 03 2026 (link)

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