# Number Sandwiches

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

## Problem

*You may wish to print off two sets of digit cards to cut out and play with.*

In this arrangement there is one number sandwiched between the "1" cards, two numbers sandwiched between the "2" cards, but only one number sandwich between the "3" cards.

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Is it possible to make a complete sandwich with one number between the "1" cards, two numbers between the "2" cards,

**and**three numbers between the "3" cards?

Click to reveal an interactivity that might help you to explore:

Click and drag the leftmost number in each pair. The number on the right will move with it.

I wonder if there is more than one way to arrange the numbers...

Can you make a complete sandwich with 1, 1, 2, 2, 3, 3, 4, 4?

Click to reveal another interactivity that might help you to explore:

I wonder if there is more than one way to arrange the numbers this time...

It is also possible to make a sandwich using 1s, 2s, 3s, 4s, 5s, 6s and 7s.

Can you find an arrangement?

Click below to see one possible solution. Is yours the same?

Image

I wonder how many different solutions there are...

*If you have any problems using the interactivities you can use the versions on the GeoGebra website: Sandwiches Interactivity.*

If you enjoyed this problem, you might like to try More Number Sandwiches.

If you enjoyed this problem, you might like to try More Number Sandwiches.

Printable NRICH Roadshow resource.

## Getting Started

Using digit cards may help you to test different arrangements.

You could start by positioning the 3s, and then see where the 1s and 2s could fit...

## Student Solutions

Amelie from Burford School, Marlow Bottom in the UK sent in a full explanation:

First, to put one number between 1, two numbers between 2, three numbers between 3, I started by putting 1 at the front. Unfortunately I couldn't find a combination that worked, so I tried putting 2 at the front. It worked...

2 3 1 2 1 3

After that I moved onto the next question: is there more than one way to do it? So I thought I've triend 1 at the front and 2 at the front, so maybe I should put 3 at the front. Yet again it worked...

3 1 2 3 2

As Amelie noticed, this is the same set of numbers backwards.

Elina from Mill Hill County High School in the UK did the opposite but still found both sandwiches:

I began by putting the largest number on the outermost side and positioned the other 2 numbers inside of the sandwich. The formation was:3,1,2,1,3,2. Then I began to see if putting the other 2 numbers (1 and 2) on the outside would also make all 3 fit in if I arranged

them correctly.

Well done also to Shriya from International School Frankfurt, Dheetchanya from Coppell ISD Middle School East in the USA, Ashlynn from ISF Academy in Hong Kong, Rishika from Nonsuch High School for Girls in the UK, Emily T, Emily H, Sam, Lyla and Hattie also from Burford School, Miss Shaw from Cockburn School in the UK, Jadon from Sacred Heart Primary Hemsworth, and Nathan from Bishop Wood C of E Junior School in the UK, who all submitted this correct solution.

Emily T had a slightly different approach:

The first problem required logic. I knew that the threes could fit around the ones' sequence. That only left two twos. I placed a two in between the ones - as the middle number. Then I could only experiment with the remaining card - a two. I tried the two in every possible place and came out with two sequences - 231213 and 312132.

The second problem required more than just logic. I used the aid of the interactivity for this problem. I knew that the ones' and twos' sequence could fit inside the fours. I tried that, but it didn't work as the ones and twos overlapped other numbers. I slid the ones beside the fours, then I put the twos in the possible places - only one worked. The gaps left were three spaces apart - perfect for the threes. I came out with the sequence - 41312432

The third problem was a bit of a mind teaser. Luckily, I had spotted a pattern - all the sequences started with the highest number, followed by a one. I put the seven and the one in these places and then I examined the sequences again . The sequences had a three in between the ones! I slotted the threes in the middle of the ones. I thought about the problem. I realised that the highest numbers would be early in the sequence, as they required more space in between them. The highest number left was six. I placed the sixes in the next possible place. Then I dealt with the fives. I slotted them in next to the sixes, but the last five and last six required the same space. So, I tried the next available space for the fives and it fitted. The next challenge was the fours. I did what I did for the last numbers. I placed the fours in the first available space - luckily it fitted. That left two spaces, two spaces apart . I slotted the only remaining cards in the remaining spaces - the twos. Finally , I had the sequence - 71316435724625.

In conclusion this problem requires patience and logic.

Emily H said:

I began with the number 2, I put both the other numbers in between the number 2's, therefore I knew that there would be two numbers between them and I could put the other pair of each number either side, so there would be one number between each 2.

Sam noticed that you can't put consecutive numbers next to each other. For example, if you have 2,1,_,_ then the second _ should be a 2 to complete the 2 sandwich but a 1 to complete the 1 sandwich.

For the sandwich with 1, 1, 2, 2, 3, 3, 4, 4, the only sandwich sent in was 4 1 3 1 2 4 3 2 (or the same sandiwch backwards). Well done to Amelie, Dheetchanya, Nathan, Ashlynn, Rishika, Emily T, Emily H, Shriya, Sam, Lyla, Hattie, Elina and Jadon, and to Lyra, Kenny and Olamide, Moyo, Grace, Ryan, Stephen and Nathan, Selah and Jordan from Monarch Global Academy in the USA who all found this sandwich.

Amelie and Elina continued by choosing which number went first. Emily T and Emily H used similar strategies to their strategies for 1, 1, 2, 2, 3, 3. Kenny and Olamide put the largest numbers in first:

To solve, start with the most troublesome number: 4. Figure out the possibilities of where the 4s an go because there are only a couple places they can go. Then fill in the blank spaces between the 4s. Do the 3s next because they're the second greatest number and there are limits to where they can go. We continued working like that until there were only two spots left for the 1s.

There were lots of possibilites for 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7. Michael S-D from Mill Hill County High School had an approach that nobody had used for 1, 1, 2, 2, 3, 3 or for 1, 1, 2, 2, 3, 3, 4, 4:

I tried to see if it would work starting with a middle ranged number, in this case 5, and then I alternated between small (S), big (B) and middle ranged (M) numbers whilst creating my solution, in the following order: M,S,B,M,S,B,M,M,M,S,B,S,B,M

My 1st solution is: 5 2 7 3 2 6 5 3 4 1 7 1 6 4 or the reverse (4 6 1 7 1 4 3 5 6 2 3 7 2 5)

My 2nd solution is: 3 5 7 4 3 6 2 5 4 2 7 1 6 1 which I found using a similar order to my last solution.

Other correct sandwiches we received were:

5 7 2 3 6 2 5 3 4 7 1 6 1 4 (Ashlynn)

4 6 1 7 1 4 5 2 6 3 2 7 5 3 (Ellees from Burford School)

1 5 1 7 3 4 6 5 3 2 4 7 2 6 (Sam)

4 5 6 7 1 4 1 5 3 6 2 7 3 2 (Hattie)

1 5 1 6 7 2 4 5 2 3 6 4 7 3 (Rishika)

1 7 1 2 6 4 2 5 3 7 4 6 3 5 (Patrick, Charlie and Alex H from Bussindale Primary School in the UK)

5 1 7 1 6 2 5 4 2 3 7 6 4 3 (Shriya and Patrick, Charlie and Alex H, Amelie)

5 7 4 1 6 1 5 4 3 7 2 6 3 2 (Patrick, Charlie and Alex H)

7 2 6 3 2 4 5 3 7 6 4 1 5 1 (Patrick, Charlie and Alex H)

7 3 6 2 5 3 2 4 7 6 5 1 4 1 (Miss Shaw, Nishant from RCHK in Hong Kong)

7 2 4 6 2 3 5 4 7 3 6 1 5 1 (Nathan)

4 6 3 5 7 4 3 2 6 5 2 1 7 1 (Amelie)

5 7 2 6 3 2 5 4 3 7 6 1 4 1 (Amelie)

Emily H, Sam and Shriya tried making number sandwiches with different combinations of numbers. Emily said:

I tried out which numbers I could use to make a complete sandwich:

I could not do it with just 1

I could not with 1 and 2

I could with 1, 2 and 3

I could with 1, 2, 3 and 4

I could not do it with 1, 2, 3, 4 and 5

I could not do it with 1, 2, 3, 4, 5 and 6,

But I could do it with1, 2, 3, 4, 5, 6 and 7!

Shriya gave some examples:

Solution with 1, 2, 4, and 5 but without 3 :

4 5 1 2 1 4 2 5

Solution with 1, 3, 4 and 5 :

3 5 4 1 3 1 4 5

solution with 1, 2, 3, 4, 5, and 7 but without 6 :

7 1 3 1 4 5 3 2 7 4 2 5

Solution with 1, 2, 3 and 6 but without 4 and 5 :

6 3 1 2 1 3 2 6

Solution with 1, 3, 4, 5,and 7 but without 2 and 6 :

4 7 5 3 1 4 1 3 5 7

More solutions are possible when you leave 1 or 2 numbers out.

## Teachers' Resources

### Why do this problem?

A hook, in the form of images, manipulatives, or interactivity, draws students in to a task as their natural curiosity compels them to explore.In this problem, a simple picture offers an intriguing challenge at several different levels, and the accompanying interactivity is engaging and enticing.

The problem is particularly valuable as it gives students an opportunity to work on a proof to explain why something is impossible (e.g. the 2-sandwich).

As there are many solutions in the case of 7-sandwiches and 8-sandwiches, the problem provides an opportunity for many students to discover their own solution, different to any that have already been found.

### Possible approach

It is helpful to have digits to rearrange. You may wish to print off two sets of Digit Cards for each group of students. Alternatively, if computers are available, students could use the interactivities.Begin by introducing the problem with digits 1 to 3:

Image

"In this arrangement there is one number sandwiched between the '1' cards, two numbers sandwiched between the '2' cards, but only one number sandwich between the '3' cards."

"Is it possible to make a complete sandwich with one number between the '1' cards, two numbers between the '2' cards, and three numbers between the '3' cards?"

Give students plenty of time to explore. When they find a solution, challenge them to try with 1-4, and then 1-7, and 1-8, recording carefully any solutions they find.

Once students have worked on the problem, invite a few students to write up their solutions on the board as a focus for discussion in a plenary. Focus on the following

**key questions:**

Can you make 2-sandwiches? How do you know?

Are any sandwiches the same looked at in different ways?

How many different 3-sandwiches are there? What about 4-sandwiches?

Did anyone manage to find a 7-sandwich? Did anyone find a different one?

What about an 8-sandwich?