Section 5 · Count Out and Transfer
The Middle Card
You choose a card and three words, and you spell each word through the packet, a card for every letter. Then you split the packet again and again, keeping the odd pile, until one card is left. It is yours.
Watch the lectureLecture 43 · The spectator finds the middle card
Try it
You choose the card and every word. The cards follow the real dealing rules, and nobody looks for your card.
Why it works
Your card went in with six cards above it and six below. That puts it at position 7 of 13, the exact centre. We never need to know its face, only its place.
Each word is one count out and transfer: deal one card per letter onto a single pile, then drop the cards still in your hand on top. Dealing onto a pile reverses the counted cards. So a card that gets counted, starting at position i, ends at position 14 − i. A card below the counted part simply moves up by the count.
Every menu word has at least seven letters: affable 7, engaging 8, persistent 10, bewildering 11. So your card is always among the counted cards, and it lands at 14 − 7 = 7. The count does not appear in that sum, which is why the words may change from round to round.
Any count of seven or more turns the packet about its centre, so position 7 never moves; each odd-pile deal then keeps that card in the middle of a smaller odd packet.
The alternate deal sends positions 1, 3, 5, … to the left and 2, 4, 6, … to the right, and each pile comes out reversed because new cards land on top. Thirteen splits into 7 and 6. Position 7 is the fourth of the seven left arrivals, so it is the middle of the odd pile. Then seven cards split 4 and 3, and the middle goes right; three split 2 and 1, and the middle goes right again. The rule is “keep the odd pile”, not “keep the left one”.
Step through the episode’s worked example below, with words of 7, 11 and 8 letters. Swap in a six-letter word at any round to watch the guarantee break.
Challenge
Repair a performance that went wrong, or rebuild the trick for a packet of a different size.
Perform it yourself
Set-up
- Any shuffled deck. Nothing is prepared.
- Write the menu where everyone can see it throughout: affable, engaging, persistent, bewildering.
- Keep a few spare cards apart, to demonstrate the dealing.
Performance
- The spectator chooses any card, remembers it and keeps its face hidden. Look away while they check it.
- Count six other cards openly onto the table, then another six. The spectator puts their card on the first group, then the second group on top. Everything stays face down: thirteen cards, the chosen one in the middle.
- Demonstrate with the spare cards: deal one card for each letter into a single pile, each new card on top; then put the cards still in your hand on top of that pile. That is one mixing move.
- Three times, the spectator chooses a menu word (the same or a different one each time), spells it silently while dealing one card per letter, and puts the remainder on top.
- Square the packet and put the menu aside. The spectator deals alternately left, right, left, right…, starting on the left, each card on top of its pile, until the hand is empty.
- Keep the pile with an odd number of cards: seven, on the left. Move the six well out of the way. They are never rejoined.
- Deal the seven the same way, starting on the left again: four and three. Keep the three, on the right. Deal again: two and one. Keep the single card, on the right.
- Ask the spectator to name the card they remembered. Then: “Turn over the survivor.”
What must not change
- Six and six. Unequal groups put the card off-centre, and nothing later can repair that.
- Every count from 7 to 13. A six-letter word sends the centre card to the top. If you change the menu or its language, count the letters again; spaces and punctuation are not dealt.
- No cuts between the rounds, and the remainder always goes on top.
- Keep the odd pile. The kept side changes: left, then right, then right. Always start each deal on the left.
- If a count or transfer goes visibly wrong, say “Let us reset the packet so we can follow the rule accurately,” rebuild it openly with the card between two groups of six, and start again.
- Rehearse with the chosen card face up, so you can watch it stay in the middle. Then perform with it face down.
Going deeper
The centre of any odd packet
Take a packet of 2m + 1 cards. Its centre is position m + 1. If a count is at least m + 1, the centre card is counted, so it goes to (2m + 2) − (m + 1) = m + 1. It does not move, whatever the count. For thirteen cards, m = 6, which is why the menu words need at least seven letters.
The lower bound is exact. With a count of six, the centre is left in the untouched remainder and moves up six places, from 7 to 1. For fifteen cards the centre is 8, so every word needs 8 to 15 letters, and affable no longer qualifies.
Why the odd pile always holds the centre
Split an odd packet by its size modulo four. With 4r + 1 cards, the left pile gets 2r + 1 and the right 2r; the centre, position 2r + 1, is odd, so it goes left as the middle arrival. With 4r + 3 cards, the left gets 2r + 2 and the right 2r + 1; the centre, position 2r + 2, is even, so it goes right, again as the middle arrival.
Reversing a pile cannot move its middle card. So each round leaves a smaller odd packet with the same card at its centre, and the rounds end at a single card. Proving survival for one round would not be enough: the invariant has to be restored, ready for the next round.
Unequal counts that cancel (episode 44)
Here the counts may differ from round to round, and each one fixes the centre on its own. Episode 44 asks a stronger question: when does a pair of different counts, s then t, repeated as s, t, s, t, restore the whole packet? Whenever s + t is at least the packet size. Split the packet into a top block of n − t cards, a middle block of s + t − n, and a bottom block of n − s. One pair reverses the top and bottom blocks and leaves the middle block exactly in place; a second pair reverses them back.
Any two menu words add up to at least fourteen, so every pair qualifies. Affable then bewildering, for instance, protects positions 3 to 7 after each pair, and the centre is always inside the protected block. With four equal counts (episode 42), the four transfers restore every card, and the centre is a fixed point: a cycle of length one.
Sources
Colm Mulcahy, Mathematical Card Magic: Fifty-Two New Effects (CRC Press, 2013): Chapter A, the Odd Location method, pp. 33–34, with the count-out-and-transfer (COAT) move and its conditions on pp. 30–39; unequal counts in Chapter 5, pp. 105–114. The word menu, the worked example and the challenges are teaching adaptations from the course scripts (episodes 41–44).