### Calendar Capers

Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?

### Reverse to Order

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

### Card Trick 2

Can you explain how this card trick works?

##### Stage: 3 Challenge Level:

Here is a very interesting observation from Abigail (Chelmondiston Primary School).
"32 doubled is 64 and 46 halved is 23."

Think about it! Can you find more examples of multiplications that work the same way?

Jasmine took a look at

32 × 46 = 1472
23 × 64 = 1472

and sent us her findings:

I have thought about these numbers and something catches my eye:

If you times the 3 from the 30 with the 4 from the 40, then you get 12.

If you times the units digit numbers (2 and 6) together, then you get 12 again.

The same thing if you times the 2 from the 20 and the 6 from the 60 and so on.

To prove my theory right, here is another example:

48 x 42 = 2016
84 x 24 = 2016

As you can see, the same thing happens here, but the number I get is 16.

Daniel (Anglo-Chinese Primary School) used some algebra to look at how the numbers relate to each other and came to the same conclusion about the 'tens' digits and the 'units' digits:

If ab x cd = ba x dc

(10a + b) (10c + d) = (10b +a) (10d + c)

100ac + 10ad + 10bc + bd = 100bd + 10bc + 10ad + ac

99ac = 99 bd

ac = bd

So 32 x 46 = 23 x 64

because 3x4 = 2x6

Daniel gave two more examples:

36 x 21 = 63 x 12 = 756

13 x 62 = 31 x 26 = 806

Does Abagail's doubling and halving idea work with these examples?