Baby Hummer
You shuffle four cards, remember the bottom one, and mix them with your own hands: cuts, and turning two cards over together. After three short instructions, exactly one card faces the other way. It is yours.
Watch the lectureLecture 04 · Baby Hummer: four cards refuse to stay mixed
Try it
Peek at a card, then do all the mixing yourself. Nobody follows your card; the cards obey the real moves.
Why it works
Number the four places from the top, 1 to 4, and picture them as seats round a tiny table. Seat 1 faces seat 3, and seat 2 faces seat 4. Call two opposite seats mates. Mates are a relationship between places, not a permanent pairing of cards.
After the set-up, three cards face one way and one card faces the other. Call that card the odd one out. Right after the set-up, your card sits in seat 3 and the odd one out in seat 1: they are mates. The whole trick protects that one fact.
A cut moves everyone round the table together. Opposite seats stay opposite and no card turns over, so the fact survives. Turning the top two together reverses their order and turns both over. If the odd one out is in that pair, the pair keeps its up–down pattern but its cards swap, so the partner takes over the odd one out’s seat. If it is not, the pair turns to match it, the fourth card becomes the new odd one out, and your card — which must be in the pair — slides over by one seat. Either way, your card and the odd one out end up opposite each other again.
The finish sounds like three unrelated instructions. Together they turn over the two cards that were in seats 1 and 3 just before the finish — one complete pair of mates — and rotate the packet, which cannot change who faces which way. If that pair is your card and the odd one out, the odd one out rejoins the crowd and your card is left alone. If it is the other pair, those two join the odd one out’s direction and your card is left alone facing its original way. That is why your card is sometimes the only face-up card and sometimes the only face-down one.
Your card always sits opposite the odd one out, and the finish turns over exactly one opposite pair — so your card is the one left facing the other way.
Step through the round from the episode: A♣, 2♥, 3♠, 4♦ face down, cuts of 3, 1 and 2 with a turn after each, and a final cut of 3. Tags above the cards are teaching labels; the gold line joins your card to the odd one out.
Challenge
Before each turn of the top two, tap the card you think will become the odd one out. Get three right to unlock the finish, and win by predicting whether your card will end face up or face down. Then break the rule — flip the top two separately — and run the finish to see what it costs.
Perform it yourself
Set-up
- Hand the spectator four different cards, face down. They may shuffle them freely.
- Turn away. They look at the bottom card and remember it, without changing its position.
- They move the top card to the bottom without turning it over.
- They turn the new top card face up and put it back on top. Exactly one card now faces up. From here on, no more free shuffling.
Performance
- Explain the two moves. A cut: any number of cards from the top to the bottom, carried together, order unchanged. A turn: take the top two together, turn that little packet over, and put it back on top. Demonstrate the turn slowly on a separate packet — the lower card comes to the top.
- The spectator cuts and turns the top two together, as many times as they like, changing the cut each time. They finish with one last straight cut.
- Give the finish one instruction at a time. First, turn the top card over, whichever way it faces, and move it to the bottom. Second, move the next top card to the bottom without turning it over. Third, turn the new top card over and leave it on top.
- Ask them to name their card. Then they spread the four cards carefully, keeping every face as it is. Three face one way; the lone exception is their card. It may be the only face-up card or the only face-down card — both count.
Turn the two together, never separately. Flipping each card where it lies does not swap them, and it breaks the trick. If they ask, say: “Keep the two together and turn the small packet over. The lower one becomes the upper one.”
To repeat, gather all four, turn them all face down, and start again from the shuffle: the starting condition is part of the trick. If a handling slip happens, say so and restart openly rather than hoping.
Going deeper
Turning a packet is two operations
Turning the top k cards over as one packet does two things at once. It reverses their order: the card in place i goes to place k + 1 − i. And it flips every face: a face-down card (0) becomes face up (1), so a bit b becomes 1 − b. Do the same turn twice and every card returns to its place and its face, so a turn is its own undo.
Flipping the two cards separately keeps the second half and loses the first. For a mixed pair the difference is stark: down–up turned together stays down–up (the cards swap), while down–up flipped separately becomes up–down.
The forbidden move never gets lucky
We checked by computer every arrangement the legal moves can reach from the set-up — 48 for each of the 24 starting orders. In every one, your card is opposite the odd one out and the finish reveals it. And if the top two are flipped separately once, at any moment, and only legal moves follow, the finish shows the wrong card in every one of the reachable states. Legal moves never repair the relationship a separate flip breaks. Only a second separate flip can, because flipping in place is its own undo.
Only a few arrangements are ever reachable
Four cards can be ordered and oriented in 4! × 24 = 384 ways. Cuts and two-card turns reach only 48 of them. Later in the course a single parity signature — position plus original label plus face, counted modulo 2 — explains exactly which ones, and why they all share the same secret.
Sources
Persi Diaconis and Ron Graham, Magical Mathematics, Chapter 1 (pp. 1–4, 11): Charles Hudson’s Baby Hummer, built on Bob Hummer’s ideas, with the set-up and finish used here. Colm Mulcahy, Mathematical Card Magic, Chapter 9 (pp. 209–212), on the cut-and-turn-two shuffle (CATO). Course episodes 04 (performance) and 05 (turning a packet is two operations).