CATO (Part 1 of 2)

By Matt Baker - Wednesday, September 9, 2020


We begin with a simple but deceptive routine, based on Charles Hudson's “Baby Hummer” trick, which is a good trick to do over Zoom since the magician never needs to touch the cards. The following description is adapted from the book Magical Mathematics by Persi Diaconis and Ron Graham (Ron unfortunately passed away just a couple of months ago — Rest In Peace, Ron). Grab a deck of cards and follow along!

Begin with a packet of four face-down cards. Look at and remember the bottom card of the packet. Now do the following:

  1. Take the top card of the packet and place it on the bottom.
  2. Turn over the top card and place it back on top.
  3. Give the packet a cut (thus moving either one, two, or three cards from top to bottom).
  4. Take the top two cards, flip them over, and place them back on top.
  5. Repeat steps 3 and 4 as many more times as you like.
  6. Take the top card, turn it over, and place it on the bottom.
  7. Take the top card and put it on the bottom.
  8. Turn over the top card and place it back on top.

Now spread the four cards: three of them will be facing the same way and the odd card out will be the one you’re merely thinking of!

The primary secret behind this trick is something called the CATO principle (not to be confused with the Libertarian institute of the same name), which was first described by the eccentric and prolific genius Bob Hummer. To see what the CATO principle is all about, begin once again with a packet of four face-down cards. Turn the 2nd and 4th cards face up, so that the cards alternate D-U-D-U (and if you say this out loud, it’s guaranteed to make small children laugh). Now do the following:

  • Take the top 2 cards, flip them over, and drop them back on top. The cards will still alternate D-U-D-U.
  • Take the top 2 cards and cut them to the bottom; the cards still alternate D-U-D-U.
  • Take the top 2 cards, flip them over, and cut them to the bottom. D-U-D-U.
  • Flip the whole packet over. D-U-D-U again! (There’s D-U-D-U as far as the eye can see...)

The invariance of the D-U-D-U pattern with respect to these operations is the essence of the basic CATO (Cut And Turn Over) Principle. (Historical note: this is also referred to in the magic literature as the CATTO “Cut And Turn Over Two” Principle and the Hummer Parity Principle.)

Bob Hummer Bob Hummer

Note also that if you cut either one or three cards from the top to the bottom, the pattern becomes U-D-U-D. It follows that if you do any combination of moves of the form (a) flip over the top 2 cards, (b) give the packet an arbitrary cut, or (c) flip over the whole packet, the cards will always alternate either D-U-D-U or U-D-U-D.

More generally, start with any packet consisting of an even number of cards which alternate D- U-D-U-...-D-U. If you do any combination of moves of the form (a) flip over an even number of cards (which might be the whole packet) or (b) give the packet a complete cut, the cards will always alternate either D-U-D-U-...-D-U or U-D-U-D-...-U-D. This is a more general and more sophisticated form of the CATO Principle.

Now, why does the Baby Hummer trick work? Well, after steps 1 and 2 the orientation of the cards is U-D-D-D, with the selection third from the top. If you were to flip the selection face-up the sequence would be U-D-U-D, which is a familiar pattern we’ve seen before. So let’s imagine from now on that the selected card has been flipped face-up. In this case, we already know that steps 3 and 4 (repeated as many times as you like, as in step 5) either maintain the U- D-U-D pattern or change it to D-U-D-U. If the pattern is U-D-U-D, steps 6, 7, and 8 will transform it into D-D-D-D (try it with cards in your hand and you’ll see). On the other hand, if the pattern is D-U-D-U, steps 6, 7, and 8 will transform it into U-U-U-U. Either way, the cards will all have the same orientation.

But wait! We only imagined turning over the selection. In reality, it will have the opposite orientation from the way it ended up in our little thought experiment. Since in our imaginary world the cards all had the same orientation, in reality the selection will be the lone card facing the other way. Boom.

***

My friend and fellow Atlanta magician Joe M. Turner recently devised the following 8-card routine based on the CATO principle. It has not been published before, and with Joe’s permission I’d like to share it with you here.

Joe’s routine, which he calls “Eight is Enough”, was inspired by Doug Canning's 6-card version of the Baby Hummer trick from Jerry Mentzer’s Card File 2. Joe’s version has the interesting feature that the trick always finishes with the selection being face-up among the face down cards, and not just oriented differently. This is a valuable addition if you’re doing this trick for a group of people over Zoom, since it makes for a consistent picture at the end — everyone gets the same result.

The Set-Up Phase:

  1. Take any 8 cards and mix them up.
  2. Remember the card at the face of the packet.
  3. Hold the packet face down in your left hand. Let's try some maneuvers to make sure you understand the instructions to come.
  4. Move one card from the top — or the middle — to the bottom. Piece of cake, right?
  5. Move another card from the top — or the middle — to the bottom, but turn it face up on the bottom. Good?
  6. Now move two cards from the top to the bottom. Then turn over just the bottom card.
  7. Push off the top two cards, turn them over as a group, and place them back on top.
  8. Turn just the top card over.

The Cutting Phase:

  1. Now cut anywhere you like and complete the cut.
  2. Turn over the top two cards as a group.
  3. Cut again, anywhere you like, and then turn over the top two cards.
  4. Now cut again and turn over the top two cards — or the top four cards. Your choice!
  5. If you'd like to turn the whole packet over, go ahead. If not, don’t!
  6. Cut anywhere once again, and then turn over either 2, or 4, or 6 cards. Or none. Or just turn the whole packet over.

The Revelation Phase:

  1. Without changing the order, spread through the cards. Do you see your card? If so, turn your packet over so that you can't see it. If you don't see your card, don’t do anything.
  2. Cut the packet so there’s a face-up card on top. This places your card face down in an unknown spot, with a face-up card on top.
  3. Turn over the top card, then the top 2 cards, then the top 3 cards, then the top 4 cards.
  4. Turn the whole packet over.
  5. Turn over the top card, then the top 2 cards, then the top 3 cards.
  6. Do you still remember your card? Say the name of your card out loud. Now spread your packet and I think you’ll find something incredible...

The spectator spreads the packet and her thought-of card is the only one face up!

Joe adds the following notes:

  • Instead of using eight playing cards, you can customize the routine to use eight index cards, or business cards, and have spectators write names, locations, companies, products, etc. on the cards.
  • Instead of cutting, you could spell various words by moving cards from the top to the bottom for each letter. Let each person choose a word or name to spell; these personal choices make it clear that everyone is not somehow cutting at the same place.
  • An advantage of using 8 cards rather than 4, as in the Baby Hummer trick, is that the extra cards make it more impossible-seeming, and they also make the “cut anywhere” steps feel more fair. The main disadvantage, of course, is that there are now more steps involved. So try to frame the set-up phase as a “practice session” to help make the trick feel shorter.

This post is already quite long, so I won’t go through all the math of why this trick works. Suffice it to say that, thanks to the CATO principle, Steps 9 through 14 can be ad libbed ad infinitum with a mixture of cuts, turning over an even number of cards, or turning over the whole packet. Emphasize the multiple options each person has which makes their path different from the ones taken by the other participants.

Credits: An extensive bibliography of Bob Hummer's own CATO-based creations, along with subsequent effects utilizing the principle, can be found here.

Max Maven provided me with the following additional historical comments. Charles Hudson's "Baby Hummer" trick was published in the December 1967 Linking Ring. The title "Baby Hummer" was meant to express the fact that prior uses of CATO -- including all of Hummer's -- were done with larger quantities of cards. Robert Neale's eight-card approach, "Number Hummer," appeared in the February 1973 Pallbearers Review, and Max Maven's own eight-card version, "Toddler," was in his 1989 book Fifth. The “rolling” procedure used in step 17 of Joe Turner’s effect is due to Max Maven, devised in the early 1980s and introduced in 1984 on the interactive video Max Maven's Mindgames. Max's procedure is also employed in the routine "Magic as Harmonic Chaos" by Juan Tamariz, from the book Verbal Magic.


Reader comments:

Gabriel

Thursday, 10 September 2020 15:22 PM - Reply to this comment

Thank you,
as always, it's great to read your thoughts!

One suggestion for the last 3 moves (6. 7. And 8.) of the original Baby Hummer with 4 cards:
You can ask to:
- turn over 3 cards
- put the top card somewhere in the middle or bottom
- turn over the top card

It seems more random to me!

Matt

Friday, 11 September 2020 00:31 AM

Great suggestion! Thanks.

Hoyt

Thursday, 10 September 2020 20:22 PM - Reply to this comment

AWESOME!!!!!! I will be using this 5* illusion

Gabriel

Friday, 11 September 2020 10:10 AM - Reply to this comment

It may be hard to remember the steps for this type of effects.

An easy solution for 6 cards is as follows:

Set-up phase:
- face down pack, bottom card memorised
- turn over 4 cards, then 3, then 2, then 1 (4-3-2-1)

Cutting phase as usual

Revelation phase:
- turn over 1 card, then 2, then 3, then 4, then 5 (1-2-3-4-5)
- shout "STOP! I feel your card stands out. It's now the only one reversed in the pack."

You can't find easier to remember! ;)

Matt

Friday, 11 September 2020 16:11 PM

Right, and this works for any even-sized packet: if there are n cards, in the set-up phase you can turn over n-2 cards, then n-3, etc. all the way down to 1, and in the revelation phase you can turn over 1 card, then 2 cards, etc. up to n-1.

Matt

Friday, 11 September 2020 22:53 PM - Reply to this comment

Tomas Blomberg pointed out that in the Baby Hummer trick, you can replace steps 6-8 with the following alternative procedure:
(a) Flip the top card over and leave it on top. Now flip the bottom card over and leave it on bottom.
(b) Flip over either the top two cards or the bottom two cards.

It's not obvious that this accomplishes the same thing as steps 6-8; indeed, if you try both procedures starting with four face-down cards, doing steps 6-8 leaves you with U-D-U-D while doing (a) and (b) leaves you with U-D-D-U. So Tomas's revelation procedure does not actually yield the same final arrangement as the traditional one, but nevertheless, it works!

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