Kaan from from FMV Erenkoy Isik School in Turkey looked at this problem step by step. He wrote out the numbers on each piece of cut strip and then wrote their totals. Here is what he sent for the first strip which Ben cut into twos starting with zero:

Numbers |
0 1 |
2 3 |
4 5 |
6 7 |
8 9 |
10 11 |
12 13 |

Totals |
1 |
5 |
9 |
13 |
17 |
21 |
25 |

This is for Miles' strip which started at 1 and was cut into twos also:

Numbers |
1 2 |
3 4 |
5 6 |
7 8 |
9 10 |
11 12 |
13 14 |

Totals |
3 |
7 |
11 |
15 |
19 |
23 |
27 |

Here are the totals for Alice's strip, starting at zero and cut into threes:

Numbers |
0 1 2 |
3 4 5 |
6 7 8 |
9 10 11 |
12 13 14 |
15 16 17 |
18 19 20 |

Totals |
3 |
12 |
21 |
30 |
39 |
48 |
57 |

Then Kaan worked out what would happen if he had a strip cut into threes but started at one:

Numbers |
1 2 3 |
4 5 6 |
7 8 9 |
10 11 12 |
13 14 15 |
16 17 18 |

Totals |
6 |
15 |
24 |
33 |
42 |
51 |

Kaan carried on like this, cutting the strips into fours, fives, sixes and sevens! Here are the totals:

Numbers |
0 1 2 3 |
4 5 6 7 |
8 9 10 11 |
12 13 14 15 |
16 17 18 19 |

Totals |
6 |
22 |
38 |
54 |
66 |

Numbers |
1 2 3 4 |
5 6 7 8 |
9 10 11 12 |
13 14 15 16 |
17 18 19 20 |

Totals |
10 |
26 |
42 |
58 |
74 |

Numbers |
0 1 2 3 4 |
5 6 7 8 9 |
10 11 12 13 14 |
15 16 17 18 19 |

Totals |
10 |
35 |
60 |
85 |

Numbers |
1 2 3 4 5 |
6 7 8 9 10 |
11 12 13 14 15 |
16 17 18 19 20 |

Totals |
15 |
40 |
65 |
90 |

Numbers |
0 1 2 3 4 5 |
6 7 8 9 10 11 |
12 13 14 15 16 17 |

Totals |
15 |
51 |
87 |

Numbers |
0 1 2 3 4 5 6 |
7 8 9 10 11 12 13 |
14 15 16 17 18 19 20 |

Totals |
21 |
70 |
119 |

Kaan then wrote:

I found that if I grouped the numbers as pairs, I found every sum increases by 4.

If I made groups of 3, I found every sum increases by 9 (3 x 3).

If I made groups of 4, I found every sum increases by 16 (4 x 4).

If I made groups of 5, I found every sum increases by 25 (5 x 5).

If I made groups of 6, I found every sum increases by 36 (6 x 6).

If I made groups of 7, I found every sum increases by 49 (7 x 7).

If I made groups of 8, I found every sum increases by 64 (8 x 8).

You have been very systematic in your working, Kaan, well done.

Ian from Cleveland High School was able to explain why this pattern occurs. He says:

If the track contains two numbers, then the difference between the nth number of the previous track and the nth number of the next track is 2. So the difference in the sums of tracks increase by 2 x 2 = 4.

Well reasoned, Ian. We can then see why for groups of three numbers, the sum increases by 3 x 3 = 9 and for fours by 4 x 4 = 16 etc.

A group from Bali International School noticed that for Ben's strip and Miles' strip, you can find the sum of each cut piece by multiplying the first number on the piece by $2$ and then adding $1$. Well spotted! Can you explain why this is always the case?

They also noticed that for Alice's strip, the sum of each cut piece can be found by multiplying the first number on the piece by $3$ and then adding $3$. Again, I wonder why this always works?

And for Winston's strip, the sum of each cut piece can be found by multiplying the first number on the piece by $5$ and then adding $10$.

What fantastic pattern identification. I'd be interested to hear from anyone who could explain why these patterns occur.

I wonder whether anybody has noticed any other kind of patterns? Do the totals for the strips cut into threes have anything else in common? Let us know if you do spot anything else by emailing primary.nrich@maths.org.