### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### It Figures

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

# Times Tables Shifts

##### Stage: 2 Challenge Level:

We had a large number of results from From Brewers Hill Middle School exploring this activity. Here are their contributions:

Paige
I looked at the 2 times table and did +1 so it looked like
2, 4, 6, 8, 10 +1 +1 +1 +1 +1 = 3, 5, 7, 9, 11
So the times table was: 2 and it shifted by: 1
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Farhan
The numbers I had received were 10, 17, 24, 31 and 38 and I figured that the times table was 7 and that the mystery number you had to add each time was 3.  For example 7 + 3=10, 14+3=17, and the pattern repeats each time.
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Lucy
I looked at the 4 times table and did +1 so it looked like:
4, 8, 12, 16, 20 +1 +1 +1 +1 +1 =  5, 9, 13, 17, 21
The times table is: 4 It was shifted by 4
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Kayleigh and Cydney
7, 11, 15, 19, 23 = 4times tables; shifted by 3
We tried out 5 first which only worked for the first numbers then we tried the 4 times tables and adding 3 which worked for the first two then worked for the rest of them when we tried it.
4n+3=7(n=1) 4n+3=11(n=2) 4n+3=15(n=3) 4n+3=19(n=4) 4n+3=23(n=5)
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Danielle
26, 40, 54, 68, 82
14 times table 12 shifted
Because if you find the difference between 26 and 40 it is 14 then take away 14 from 26 and it is 12 and that is what is shifted.
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Callie and Muna
Our solution to Times Table Shift for two level 1 questions are:
1. -15 25 35 45 55     Table:10 Shifted by:5
2. -5 8 11 14 17      Table:3   Shifted by:1  These were easy!!!!!!!!!!!!:-)
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Eddie
To get the solution you have to find the difference between the second and third number e.g. 39 and 56 is 17 so that would be your table.
Now to get the shifted answer you have to the difference between your table and the first number wich is 22 so 5 would be the shifted number. The answer would be Table 17 and shifted by 5.
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Jasmine
First I started off with the numbers 6, 10, 14, 18 and 22. I worked this problem out by starting with my 2 times table and I did 2+4=6 and then I tried to do 4+4=8 but it didn’t come up with the right answer so I tried my 4 times table and it turned out to be right, 4+2=6, 8+2=10, 12+2=14 and so on.
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Georgia
My solution for the times table shifts is, well for example get 11, 18, 25, 32, 39.
You find out how much is between 11 and 18 which is 7 so the table is 7 and then you find out how much it is from 7 till 11 and that’s 4 so that’s my solution
I can do this solution with many numbers and it works it properly.
I found them quite strange at the start but then I found my solution and I got through them so quickly.
I did Level 1 and Level 2.  The easiest one I did was level 1 because the numbers are not high but Level 2 is also as easy as Level 1. My maths set have enjoyed doing this NRICH task on your website thank you. :)
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Harry
If I had the numbers 3, 5, 7, 9 they are in the 2 times table it's the 2 I use.  How many more 1's is it to get to the first number = 1, 2+1= 3 (the first number in the sequence.)  It works on every one of the questions
Here is another example.  11, 18, 25, 32, 39  these are the 2 times table
The first number is 11  how many more 1's to get to 11 from 7=4 4 is how many shifted.
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Kai & Max
Our solution to this investigation (Times Tables Shifts) is:  work out whether it is odd or even, if it is even, do the following,  (your numbers are)    4  6  8  10  12  get the first even number above zero the first even number is 2  what do you add to 2, to get to 4?      The answer that you add to 2 is 2 (you add 2+2 to get to 4)  then get the next number in the two times table and see if it works for all of them.  2n+2=4 2n+2=6 2n+2=8 2n+2=10 2n+2=12
Now you enter the times table into the answer box (on this question it would be 2) then add in the equation into the 'shifted by' box and press check  here are some of the other equations we solved:
Numbers: 7  12  17  22  27  solution  5n+2=7 5n+2=12 5n+2=17 5n+2=22 5n+2=27
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John
On level 1 and 2 they work the same way. At the start say you had 14 26 38 50 62, you would subtract 14 from 26 which would equal 12. 12 +?= 14. So you would have to subtract 12 from 14 which is 2 , and 12 add 2 equals 14.  Which means the table is 12 and the shift is 2.
On level 3 it's quite harder as there are many problems with not all the same solutions. Here is one of the solutions. Say you had 170 161 89 62 71, you would subtract 161 from 170 which would be 9. Then you would cancel down 170 which would be 17 . 9 + ?= 17 so you subtract 9 from 17 which would be 8.  So the table would be 9 and the shift would be 8.

Thank you for your work on this, Brewers Hill, it looks as if you spent quite a while on this exploration. Well Done!  I am particularly impressed by those of you who found a way to work out the solution no matter what the numbers were.

Simran from Lincroft Middle School sent us a very clear solution:

First, what you need to do is count the number between the given numbers to find the times table. You can check this by counting to the next mumber in the sequence and checking if this is the same.

Next, start at the begining number and count backwards until you arrive at the times table you have found.  This number is your shift.

For level 3, because the numbers are jumbled, you have to put them in order from smallest to biggest. You then find the difference between these numbers by counting. The number that is the difference the most often is your times table.

Then, you need to count backwards from the smallest number you have until you get to the first number that is smaller than your times table.  This is your shift.

Ellen from Pigeon Mountain Primary in New Zealand agrees with Simran.  She summarised the way to solve this challenging by saying:

This is for Level 1 and 2 only

To find out the table, you select any two numbers beside each other on the screen. Then, you take the smaller number away from the bigger one. The answer is the table. E.g: 7, 12, 17, 22, 27: 17-12=5. 5 is the table.

The shift is simply the difference between the first number in the list and the table. E.g: 7, 12, 17, 22, 27: 17-12=5. % is the table. 7-5=2. 2 is the shift.

Well done too to Janek and Anna from Risdene Academy, and Martha from Langtons Junior Academy, who explained what to do in a very similar way to Simran and Ellen.