Challenge Level

Well done to everybody who had a go at this investigation. We were sent lots of solutions describing patterns in the house number sums, but only a few children had a go at explaining why these patterns happened. Have a look through these solutions and see if you can think of the reasons behind each of the patterns.

Ephraim and Felix from Pierrepont Gamston Primary School in England sent us their ideas. Felix spotted a pattern to do with odd and even numbers:

When you add the house numbers from side to side in a straight line you will always end up with an odd number. If you add them diagonally such as 1+4, you get 5 and then if you add the next two numbers, 2+3, you also get 5.

Well done Felix, we had a few solutions describing patterns like the one you spotted. I wonder why the total is always an odd number when you add diagonally?

Ephraim found a different pattern that occurs when you add house numbers along the same side of the road. He said:

On the first example of grouping I realised when that when you keep adding the numbers on each side the totals of the even numbers will be larger than the odd side progressively by 2 then 3 then 4 etc. With that information you will be able to predict the difference in each side no matter how far you go.

Well spotted, Ephraim. Ephraim is saying that if you add the first two houses on the odd side, the total is 4, but adding the first two houses on the even side gives a total of 6. Adding the first three houses on the odd side gives a total of 9, whereas adding the first three houses on the even side gives a total of 12. I wonder if there's a way to predict what the difference between the two sides will be when you add the first 10 houses? Or the first 100?

Lots of children from Kingswood College in Australia sent in their ideas. Hope described some patterns that she noticed:

I discovered that when the numbers are added diagonally there is a pattern because the answer is repeated. (E.g. 1 + 4 = 5, 3 + 2 = 5; 5 + 8 = 13, 7 + 6 = 13). The pattern of diagonally opposite sums will be 5, 5, 13, 13, 21, 21, 29, 29, 37, 37, 45, 45, 53, 53).

I then noticed another pattern, which is that there will be a difference of 8 between each of those answers. But I can't work out why at this stage.

Well spotted, Hope! Have a careful think about why the difference might be 8. Can you see how 1 + 4 is different from 5 + 8? How is 5 + 8 different from 9 + 12?

Aiden also spotted a pattern that occurs when we add the numbers diagonally:

I investigated adding the top first number to the bottom fifth number (1+10), (3+8), (5+6), (7+4) these answers will always be 11.

Then I continued this pattern by adding (11+20), (13+18), (15+16), (17+14), (19+12) these answers all add up to 31.

I noticed that both these numbers have 1 in the units column and a difference of 20.

Well spotted, Aiden, and well done for noticing that the answers are similar. Have a careful think about why 1 + 10, 3 + 8, 5 + 6 and 7 + 4 equal 11, whereas 11 + 20, 13 + 18, 15 + 16 and 17 + 14 equal 31. Can you see any patterns in these sums? What do you think each sum in your next set of numbers will equal?

Ryan and Bowen also noticed that the sums increase by the same amount each time:

Example A

I added the street numbers in a star shape. I discovered that the first three digit pairs all added up to 7, the next three digit pairs all added up to 19, the next three digit pairs all added up to 31, the next three digit pairs added up to 43. I wrote the sequence down like this 7, 19, 31, 43, 55, then I noticed a pattern and that there was a difference of 12 between each number.

Example B

I also added in a star shape but I used an overlapping star shape. My first three digit pairs added up to 7, then 15, then 23, then 31, then 39, then 47, then 55. The sequence looked like this 7, 15, 23, 31, 39, 47, 55. I noticed there was a pattern because there was a difference of 8 in the sequence.

Well done, Ryan and Bowen, for working out the patterns in these sequences. Have a look at each set of sums - can you see why they increase by 12 each time in Example A, and by 8 each time in Example B?

Lucas chose to add numbers on the same side of the street together as in Ephraim's solution above, but in a slightly different way. He said:

I investigated adding the first number to the last number on each side of the street. Eg 1+29, 3+27, 5+25 and so on. A pattern I noticed was that these pairs always added up to 30. However 15 doesn't have pair but if I double the number it will add up to 30. I did the same with the opposite side of the street, the even numbers. So I added 2+30, 4+ 28, 6+26 and so on. A pattern I noticed was that these pairs always added up to 32. However this time 16 doesn't have a pair but if I double it the answer will also be 32.

Well spotted, Lucas! Do you notice anything about the fact that the total of each pair on the odd side of the street is 30, and the odd one out is 15? Or that the total of each pair on the even side is 32, but the odd one out is 16? I wonder what these numbers have in common?

Ellie noticed some similar numbers when she added diagonally. Ellie said:

I added the numbers diagonally such as 1 + 30, 3 + 28, 5 + 26, 7 + 24 each answer is always 31. When I added the opposite numbers diagonally 2 + 20, 4 + 27, 6 + 25 each answer is also always 31. I think this happens because the bottom line of numbers is decreasing and the top line is increasing and that keeps the answers even. I noticed that the street numbers I used go up to 30 and my answer is always 1 more than that.

These are some good ideas, Ellie, and well done for explaining why the pattern works. I wonder if you could use this pattern to find a quick way of adding all the numbers from 1 to 30 together? Or all the numbers from 1 to 100?