# ACE, TWO, THREE...

Can you picture how to order the cards to reproduce Charlie's card trick for yourself?

## Problem

Take a look at this video.

**Can you work out how Charlie ordered the cards to perform the trick?**

Once you've had a chance to think about it, click below to see some suggested starting points:**Charlie started by thinking:**

“Ace” has three letters so it should go in third place. “Two” has three letters so it should go in sixth position. “Three” has five letters so it should go in eleventh position. “Four” has four letters so it should go in fifteenth position...

**Luke** **started by thinking:**

I should be able to work backwards, so I'll start with just the Jack, Queen and King and see what happens...

**Alison** **started by thinking:**

If I arrange the cards from Ace to King to do the trick, I won't reveal the cards in the right order. But I could keep a record of the sequence they come out in...

**Can you take each of their starting ideas and develop them into a solution?**

Can you use each method to perform the trick in a different language, or with two suits of cards together, or in reverse order from King to Ace, or...?*You may be interested in the other problems in our Seeing mathematically Feature.*

## Getting Started

It might help to have paper and pencil handy as you work through the three methods.

## Student Solutions

Lots of people sent in correct solutions to this card problem. Many people developed Charlie's thoughts into a full solution. Well done to James from Chase Terrace Technology College, Evan from Muhlenberg Middle
School, Barnaby from Devonshire Primary School, Daisy, Bobby, Molly, Bethany, Anya and
Rosie from All Saints CofE Junior School Hessle and Richard from Worplesdon
School who did this! Here is their solution:

To represent the unknown cards write thirteen places on the page where the values will go:

_ _ _ _ _ _ _ _ _ _ _ _ _

Now count down three places for A-C-E and write 'A' for Ace in the third place, because this is where Charlie showed the Ace to be.

_ _ A _ _ _ _ _ _ _ _ _ _

Keep counting T-W-O (three places) for 2, T-H-R-E-E (five places)

_ _ A _ _ 2 _ _ _ _ 3 _ _

Now to count four places for F-O-U-R we need to go back to the beginning when we reach the end, this is because Charlie puts the cards he counts past on the bottom of the pack.

_ 4 A _ _ 2 _ _ _ _ 3 _ _

And for counting out F-I-V-E, skip the places already occupied by card values; this is because Charlie removes the cards when he finds them, they're not counted again.

_ 4 A _ _ 2 _ 5 _ _ 3 _ _

If you keep counting out the names of the cards you eventually get to the answer:

Q 4 A 8 K 2 7 5 10 J 3 6 9

Fantastic! Esperanza from Hamburg, Kendyl from Kelly Elementary School Wyoming, Michael from Cloverdale Catholic School Canada, Tayla from Farm Cove Intermediate and Milly from Grasmere Primary School found the correct answer with similar methods, and some people used letters to represent the missing card values - this is a great way to do it too.

Reiss from Burton Borough School had the excellent idea of working backwards from King to Ace. Here is his solution:

Have a group of cards on the table, at the bottom is an "Ace" all in order up to a "King" which should be at the top. Start off by putting the top card of the ordered pile in your hand and then pick up the second one "Queen" then put it on top of the "King" in your hand. Spelling "Queen" is important. The Queen is a "Q" then take the bottom card in your hand then place it above the top one in your hand and that is the "U." Keep doing that until you have completed the spelling "Queen." Then you pick up the next card in the ordered pile "Jack" and place it on top of the cards in your hand and do the same again and do the same again until you have completed the spelling "Jack." Do the same thing until all of the cards in the ordered pile are in your hand. Make sure you have done it using the correct spelling of each number, otherwise it might go wrong when you come to do it.

Can you see why his method works?

Patrick and H&H also submitted solutions for saying the card names in French and German. Brilliant. Can you see why this trick will work in any language provided you order the cards correctly? Try to find the right order for a language of your choice!

To represent the unknown cards write thirteen places on the page where the values will go:

_ _ _ _ _ _ _ _ _ _ _ _ _

Now count down three places for A-C-E and write 'A' for Ace in the third place, because this is where Charlie showed the Ace to be.

_ _ A _ _ _ _ _ _ _ _ _ _

Keep counting T-W-O (three places) for 2, T-H-R-E-E (five places)

_ _ A _ _ 2 _ _ _ _ 3 _ _

Now to count four places for F-O-U-R we need to go back to the beginning when we reach the end, this is because Charlie puts the cards he counts past on the bottom of the pack.

_ 4 A _ _ 2 _ _ _ _ 3 _ _

And for counting out F-I-V-E, skip the places already occupied by card values; this is because Charlie removes the cards when he finds them, they're not counted again.

_ 4 A _ _ 2 _ 5 _ _ 3 _ _

If you keep counting out the names of the cards you eventually get to the answer:

Q 4 A 8 K 2 7 5 10 J 3 6 9

Fantastic! Esperanza from Hamburg, Kendyl from Kelly Elementary School Wyoming, Michael from Cloverdale Catholic School Canada, Tayla from Farm Cove Intermediate and Milly from Grasmere Primary School found the correct answer with similar methods, and some people used letters to represent the missing card values - this is a great way to do it too.

Reiss from Burton Borough School had the excellent idea of working backwards from King to Ace. Here is his solution:

Have a group of cards on the table, at the bottom is an "Ace" all in order up to a "King" which should be at the top. Start off by putting the top card of the ordered pile in your hand and then pick up the second one "Queen" then put it on top of the "King" in your hand. Spelling "Queen" is important. The Queen is a "Q" then take the bottom card in your hand then place it above the top one in your hand and that is the "U." Keep doing that until you have completed the spelling "Queen." Then you pick up the next card in the ordered pile "Jack" and place it on top of the cards in your hand and do the same again and do the same again until you have completed the spelling "Jack." Do the same thing until all of the cards in the ordered pile are in your hand. Make sure you have done it using the correct spelling of each number, otherwise it might go wrong when you come to do it.

Can you see why his method works?

Patrick and H&H also submitted solutions for saying the card names in French and German. Brilliant. Can you see why this trick will work in any language provided you order the cards correctly? Try to find the right order for a language of your choice!

## Teachers' Resources

Why do this problem?

This problem challenges students to visualise what is going on as they figure out how the trick was done. Students' natural frustration in wanting to know how the trick is done may provide an opportunity for you as a teacher to let them struggle for longer than usual.

At the end of the task, once the vast majority of students have succeeded, you have an opportunity to celebrate their willingness to persevere and draw attention to the importance of resilience as a characteristic of good mathematicians.

### Possible approach

Perform the trick for the class (or show the video). Hand out packs of cards so that each pair has one suit, and challenge students to work out how to order the cards to perform the trick.

Here is a worksheet with the three starting points from the problem.

Challenge students to figure out how each method works.

Finish the task by giving students a chance to explain and demonstrate each method.

Key questions

For Charlie's method: whereabouts must the Ace have been at the start of the trick?

For Luke's method: does it help to work backwards?

For Alison's method: why does the 3 come out first?

Possible extension

Use each method to work out the order needed to perform the trick in another language, or in reverse order King to Ace, or with two suits.

For more problems on visualising, see our Visualising collection.

Possible support

Charlie's method, together with paper and pencil for recording, is the most accessible.