More Number Sandwiches
When is it impossible to make number sandwiches?
Age
11 to 16
Challenge level
Problem
In Number Sandwiches you may have made sandwiches with the numbers 1 to 3, 1 to 4 and 1 to 7.
Can you use the interactivity below to make sandwiches with the numbers 1 to 5? Or 1 to 6?
Sometimes it is difficult to tell whether a task is impossible, or just very difficult!
Can you convince yourself that it is impossible to make sandwiches with the numbers 1 to 5, and 1 to 6?
Click below to reveal some questions that might help you to explain what is happening:
In a "7-sandwich", how many red squares are covered and how many blue squares are covered?
If it were possible to make a "6-sandwich", how many red squares and how many blue squares would be covered?
Click below to reveal some more questions that might help you develop your thinking further:
If you place a 1 on a blue square, on which colour will you place the other 1?
If you place a 2 on a blue square, on which colour will you place the other 2?
If you place a 3 on a blue square, on which colour will you place the other 3?...
In general, what can you say about the colours on which you place pairs of numbers?
When you try to make a sandwich with the numbers from 1 to 5, or from 1 to 6, what goes wrong?
Which other sandwiches are impossible? How can you be sure?
Student Solutions
Thank you to Ashlynn from ISF in Hong Kong, who sent in a full solution. This is Ashlynn's work:
Image
In a "7- sandwich", how many red squares are covered and how many blue squares are
covered?
There are 7 red squares covered and 7 blue squares covered.
If it were possible to make a "6 -sandwich", how many red squares and how many blue
squares would be covered?
For a 6 -sandwich, total there are 12 squares covered. 6 are red, 6 are blue.
If you place a 1 on a blue square, on which colour will you place the other 1?
Blue
If you place a 2 on a blue square, on which colour will you place the other 2?
Red
If you place a 3 on a blue square, on which colour will you place the other 3?...
Blue
In general, what can you say about the colours on which you place pairs of numbers?
If it is an odd number, it covers the same colour of tile. e.g. Red -Red, Blue- Blue
If it is an even number, it covers different colours of tile. e.g. Red -Blue, Blue- Red
When you try to make a sandwich with the numbers from 1 to 5, or from 1 to 6, what goes wrong?
For a 6 -sandwich, in total there should be 12 tiles covered. 6 are red tiles, 6 are blue tiles.
We try:
| Number | Possible Colours of Tiles Covered |
|---|---|
| 1 | R-R |
| 2 | R-B |
| 3 | R-R |
| 4 | R-B |
| 5 | B-B |
| 6 | IMPOSSIBLE! |
It is impossible because in 6 -sandwich there are 3 even numbers which cover 3R and 3B.
Odd numbers can only cover an even number of R or B. They cannot make another 3R or 3B.
For a 5- sandwich, in total there should be 10 tiles covered. 5 are red tiles, 5 are blue tiles.
We try:
| Number | Possible Colours of Tiles Covered |
|---|---|
| 1 | R-R |
| 2 | R-B |
| 3 | B-B |
| 4 | R-B |
| 5 | IMPOSSIBLE! |
It is impossible because in 5 -sandwich there are 2 even numbers which cover 2R and 2B.
Odd numbers can only cover an even number of R or B. They cannot make another 3R or 3B.
Which other sandwiches are impossible? How can you be sure?
Odd numbers can only cover an even number of R or B.
| Number Sandwich | Number of Red and Blue Need to be Covered |
Even numbers in the sandwich |
No. of R and B covered by even number |
No. of R and B needed to be covered by odd numbers |
POSSIBLE? |
| 1 | NO | ||||
| 2 | 2R, 2B | 2 | 1R, 1B | 1R, 1B | NO |
| 3 | 3R, 3B | 2 | 1R, 1B | 2R, 2B | YES |
| 4 | 4R, 4B | 2, 4 | 2R, 2B | 2R, 2B | YES |
| 5 | 5R, 5B | 2, 4 | 2R, 2B | 3R, 3B | NO |
| 6 | 6R, 6B | 2, 4, 6 | 3R, 3B | 3R, 3B | NO |
| 7 | 7R, 7B | 2, 4, 6 | 3R, 3B | 4R, 4B | YES |
| 8 | 8R, 8B | 2, 4, 6, 8 | 4R, 4B | 4R, 4B | YES |
| 9 | 9R, 9B | 2, 4, 6, 8 | 4R, 4B | 5R, 5B | NO |
| 10 | 10R, 10B | 2, 4, 6, 8, 10 | 5R, 5B | 5R, 5B | NO |
| 11 | 11R, 11B | 2, 4, 6, 8, 10 | 5R, 5B | 6R, 6B | YES |
| 12 | 12R, 12B | 2, 4, 6, 8, 10, 12 | 6R, 6B | 6R, 6B | YES |
| 13 | 13R, 13B | 2, 4, 6, 8, 10, 12 | 6R, 6B | 7R, 7B | NO |
| 14 | 14R, 14B | 2, 4, 6, 8, 10, 12, 14 | 7R, 7B | 7R, 7B | NO |
| 15 | 15R, 15B | 2, 4, 6, 8, 10, 12, 14 | 7R, 7B | 8R, 8B | YES |
We can see a pattern:
If the number sandwich is an even number. When divided by 2, if the quotient is odd number, then it is impossible to make the number sandwich.
If the number sandwich is an odd number. When [decreased] by 1 and then divided by 2, if the quotient is odd number, then it is impossible to make the number sandwich.
Teachers' Resources
Why do this problem?
This problem follows on from Number Sandwiches for those students whose curiosity has been sparked by the initial problem and who are desperate to know "What if...?"Having explored 3-, 4- and 7-sandwiches, the natural question that arises is "Can I make 5- and 6-sandwiches?"
Possible approach
Once students have had plenty of time to work on the original problem, some students will perhaps wonder why you have not asked them to explore 5- and 6-sandwiches. The interactivity is coloured red and blue to draw students' attention to what happens when you place a pair of odd numbers, or even numbers, on the grid.We hope that this, together with the key questions below, will help students to construct a proof for the impossibility of 5- and 6- sandwiches, and perhaps generalise to the other impossible sandwiches.
You can see an outline of the proof in the article Impossible Sandwiches
Key questions
In a "7-sandwich", how many red squares are covered and how many blue squares are covered?If it were possible to make a "6-sandwich", how many red squares and how many blue squares would be covered?
If you place a 1 on a blue square, on which colour will you place the other 1?
If you place a 2 on a blue square, on which colour will you place the other 2?
If you place a 3 on a blue square, on which colour will you place the other 3?...
In general, what can you say about the colours on which you place pairs of numbers?
When you try to make a sandwich with the numbers from 1 to 5, or from 1 to 6, what goes wrong?