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Summing Consecutive Numbers

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Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Your Number Is...

Age 7 to 14 Challenge Level:

Leo, Mikki, Jodi, Amy, Cienna, Jedd, Charlie, Freya, Orla, Freya, Merry, Spike and Mo from Reepham Primary School worked on this task. Here is one of their submissions:
This problem was an algorithm. This works because when doing the calculation you are working out the opposite and the middle part in the calculation always equals five. Also we worked out that doing the calculations backwards and using the opposite symbol also gives you five. This proves that you can do it in any order.

For example:
10 x 2 = 20
20 - 4 = 16
16 $\div$ 2 = 8
8 - 3 = 5

Shao from The Australian International School in Malaysia
 wrote:
You think of a starting number. For example, 45. When you add 3, you have your original number plus 3. In this case, 48.
When you double it, you have twice your original number plus 6. In this case, 96.
Then you add 4, and now you have twice the original number plus 10. In this case, 100.
Then you halve your number, and you will have your original number plus 5. In this case, 50.
Finally, when you subtract the original number, you will have:
Your original number - Your original number + 5 (which is obviously, 5!)

This is always true whatever number you chose at the beginning because at the end, you’re subtracting your original number from your original number. And since whenever you subtract a number from itself, you always get 0.

For example, if we use 99 as the original number, we will still end up with:
Your original number - Your original number + 5 = 0 + 5 = 5.


Fergus from Golspie High School in Scotland explained this idea very simply:
At the end you subtract the original number so if you ignore the original it makes it :
3, $\times$ 2 = 6,  plus 4 = 10  then halved is 5


Aiman added the folowing:-
When you enter a number and follow all the instructions, you always get 5 at the end.
The idea is that, if (this is an example) 3 was added at the start to another random number, the number and 3 seem to be added together, but by subtracting the other number at the end, the answer will always be 3.

If you look at the blue line and orange dots that are next to the instructions that the interactivity gives, the blue line (a random number) and the orange dots (the units that the interactive tells a person to add) are separate, even though when the instructions are used, it is one number that a person sees.

While the instructions are performed, the answer from the second last step is (random number + 5). You can see that even though the number goes through a lot of equations, you still end up with (random number + 5).

The next step subtracts the random number, leaving you with 5. Whatever number is chosen, the answer will always be 5. 

Anh Minh from British Vietnamese International School Hanoi, Steven from Reading School in the UK and Qianwei from Humanitree in Mexico showed how this works using algebra. This is Anh Minh's work:
Start with $x$
Add $3:$ $x+3$
Double: $2(x+3) = 2x+6$
Add $4:$ $2x +6+4= 2x + 10$
Half: $\frac12(2x+10) = x +5$
subtract first number: $x+5-x= x-x+5= 5$
Answer always be $5$


Elifsu from International School Laren in Holland made some 'think of a number' machines in a fully animated powerpoint presentation. Click here to download the presentation. 

Shreya, Sirat, Lelia, Divya, Kirsten, Reetinderit, Ryan, Harman, Shazana and Agam from ​Glendenning Public School in Australia also worked on this task. Here are two of their solutions:

For this challenge task, I figured out that the rule given worked for any number that I tried. It also worked for the numbers 0 and 1. The result was always the number 5. I figured out that this might be because if I removed the first and last instruction of the given rule, the answer was always 5.

I also figured out two more patterns that had a similar rule. The first pattern that I worked out required me to change the numbers of the rule slightly:
Instead of 'adding 3 and 4' to your chosen number, I decided to change it and make it so that you would have to 'add 2 and 2' instead. My pattern looked something like this:

Think of a number: 10    
Add 2: 12    
Double: 24    
Add 2: 26    
Halve: 13
Take away the number you started with: 3

I continued this pattern with a variety of numbers, I tried nearly all of the numbers from 0 to 10. The result for all of the numbers I tried with was 3.       

AND

Instead of always getting 5 in our method you could always get the number 9 once you finish.
Here is the method we discovered...
Think of a number    
Add 10     
Subtract 5    
Add 4
And subtract the first number.

We tried this method with a majority of 1, 2 and 3 digit numbers including... 4
Here is what we got...
4 + 10 = 14
14 - 5 = 9
9 + 4 = 13
13 - 4 = 9

For 23 we got...   
23 + 10 = 33
33 - 5 = 28
28 + 4 = 32
32 - 23 = 9

For 101 we got...   
101 + 10 = 111
111 - 5 = 106
106 + 4 = 110
110 - 101 = 9

We then tried many other one-, two- and three-digit numbers, and then came to a conclusion that when you finish the method it will always equal 9.

Thank you all for these submissions.