Sunday Times Teaser 3273 – A good Deal?

by BRG on 15 June 2025

by Victor Bryant

Published Sunday June 15 (link)

Delia plays a game with a stack of different pieces of card. To shuffle the stack she deals them quickly into four equal piles (face-downwards throughout) — the top card starts the first pile, the next card starts the second pile, then the third starts the third pile and likewise the fourth. She continues to place the cards on the four piles in order. Then she puts the four piles back into one big stack with the first pile on the bottom, the second pile next, then the third, with the fourth pile on the top.

She is very pleased with her shuffling method because each card moves to a different position in the stack, so she decides to repeat the shuffle a few more times. However, this is counter-productive because after fewer than six such shuffles the cards will be back in their original order!

How many cards are there in her stack?

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BRG permalink


# shuffle a pack of cards ussing <p> piles
def shuffle(pack, p=4):
  s = tuple(pack[i::4][::-1] for i in range(p))
  return sum((s[i] for i in reversed(range(p))), [])

# consider the number of cards in the pack
for nc in range(4, 100, 4):
  # create the pack
  pack = list(range(nc))
  # shuffle it once ...
  once = shuffle(pack)
  # ... and check that all cards have moved
  if any(p == q for p, q in zip(pack, once)):
    continue
  # count shuffles (< 6) that return the pack to its original state
  shf_cnt = 1
  while shf_cnt < 6 and any(p != q for p, q in zip(pack, once)):
    once = shuffle(once)
    shf_cnt += 1
  # there must be 'a few more' shuffles after the first
  if 2 < shf_cnt < 6:
    print(f"The pack contains {nc} cards ({shf_cnt} shuffles).")
    break

# shuffle a pack of cards ussing <p> piles

def shuffle(pack, p=4):

s = tuple(pack[i::4][::-1] for i in range(p))

return sum((s[i] for i in reversed(range(p))), [])

# consider the number of cards in the pack

for nc in range(4, 100, 4):

# create the pack

pack = list(range(nc))

# shuffle it once ...

once = shuffle(pack)

# ... and check that all cards have moved

if any(p == q for p, q in zip(pack, once)):

continue

# count shuffles (< 6) that return the pack to its original state

shf_cnt = 1

while shf_cnt < 6 and any(p != q for p, q in zip(pack, once)):

once = shuffle(once)

shf_cnt += 1

# there must be 'a few more' shuffles after the first

if 2 < shf_cnt < 6:

print(f"The pack contains {nc} cards ({shf_cnt} shuffles).")

break

John Z permalink


# consider number of cards up to 100 (except trivial case of four cards)
for nc in range(8, 101, 4):
  deck = tuple(range(nc))

  # pile size
  ps = nc // 4
  
  # up to five shuffles
  for i in range(5):
    # shuffle cards by zipping cards from n groups of 4 where n = nc // 4 
    deck = sum(zip(*(deck[i * 4: (i + 1) * 4] for i in range(ps))), ())[::-1]
    
    # check all cards change position for this number of cards
    if i == 0 and any(x == y for x, y in zip(deck, range(nc))):
      break
    
    # check that the latest shuffle gives the original deck
    if deck == tuple(range(nc)):
      print("Number of cards =", nc, "; number of shuffles:", i + 1)

# consider number of cards up to 100 (except trivial case of four cards)

for nc in range(8, 101, 4):

deck = tuple(range(nc))

# pile size

ps = nc // 4

# up to five shuffles

for i in range(5):

# shuffle cards by zipping cards from n groups of 4 where n = nc // 4

deck = sum(zip(*(deck[i * 4: (i + 1) * 4] for i in range(ps))), ())[::-1]

# check all cards change position for this number of cards

if i == 0 and any(x == y for x, y in zip(deck, range(nc))):

break

# check that the latest shuffle gives the original deck

if deck == tuple(range(nc)):

print("Number of cards =", nc, "; number of shuffles:", i + 1)

John Z permalink

Curiously, a radically different shuffle technique gives the same result; ie nc // 4 piles of 4;=:


# consider number of cards up to 100 (except trivial case of four cards)
for nc in range(8, 101, 4):
  deck = tuple(range(nc))

  # pile size
  ps = nc // 4
  
  # up to five shuffles
  for i in range(5):
    # shuffle cards by zipping cards from n groups of 4 where n = nc // 4
    #deck = sum(zip(*(deck[i * 4: (i + 1) * 4] for i in range(ps))), ())[::-1]
    # shuffle cards by zipping cards from four groups of nc // 4 cards
    deck = sum(zip(*(deck[i * ps: (i + 1) * ps] for i in range(4))), ())[::-1]

    # check all cards change position for this number of cards
    if i == 0 and any(x == y for x, y in zip(deck, range(nc))):
      break
    
    # check that the latest shuffle gives the original deck
    if deck == tuple(range(nc)):
      print("Number of cards =", nc, "; number of shuffles:", i + 1)

# consider number of cards up to 100 (except trivial case of four cards)

for nc in range(8, 101, 4):

deck = tuple(range(nc))

# pile size

ps = nc // 4

# up to five shuffles

for i in range(5):

# shuffle cards by zipping cards from n groups of 4 where n = nc // 4

#deck = sum(zip(*(deck[i * 4: (i + 1) * 4] for i in range(ps))), ())[::-1]

# shuffle cards by zipping cards from four groups of nc // 4 cards

deck = sum(zip(*(deck[i * ps: (i + 1) * ps] for i in range(4))), ())[::-1]

# check all cards change position for this number of cards

if i == 0 and any(x == y for x, y in zip(deck, range(nc))):

break

# check that the latest shuffle gives the original deck

if deck == tuple(range(nc)):

print("Number of cards =", nc, "; number of shuffles:", i + 1)

John Z permalink

A slightly different approach:


# consider number of cards up to 100 (except trivial case of four cards)
for nc in range(8, 101, 4):
  # pile size
  ps = nc // 4
  
  deck = tuple(range(nc))

  # create shuffle map giving new position for each card after shuffle 
  map = sum(zip(*(deck[i * 4: (i + 1) * 4] for i in range(ps))), ())[::-1]
  
  # up to five shuffles
  for i in range(5):
    # shuffle cards by moving each card to a new position determined by  
    deck = tuple(deck[t] for t in map)
    
    # check all cards change position for this number of cards
    if i == 0 and any(x == y for x, y in zip(deck, range(nc))):
      break
    
    # check that the latest shuffle gives the original deck
    if deck == tuple(range(nc)):
      print("Number of cards =", nc, "; number of shuffles:", i + 1)

# consider number of cards up to 100 (except trivial case of four cards)

for nc in range(8, 101, 4):

# pile size

ps = nc // 4

deck = tuple(range(nc))

# create shuffle map giving new position for each card after shuffle

map = sum(zip(*(deck[i * 4: (i + 1) * 4] for i in range(ps))), ())[::-1]

# up to five shuffles

for i in range(5):

# shuffle cards by moving each card to a new position determined by

deck = tuple(deck[t] for t in map)

# check all cards change position for this number of cards

if i == 0 and any(x == y for x, y in zip(deck, range(nc))):

break

# check that the latest shuffle gives the original deck

if deck == tuple(range(nc)):

print("Number of cards =", nc, "; number of shuffles:", i + 1)

Sunday Times Teaser 3273 – A good Deal?

by Victor Bryant

Published Sunday June 15 (link)

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