32×38 = 30×40 + 2×8 34×36 = 30×40 + 4×6 56×54 = 50×60 + 6×4 73×77 = 70×80 + 3×7

Chloe and Nina (Y7) St James Middle School, Bury St Edmunds found these examples:

81×89 = 80×90 + 1×9 = 7209 72×78 = 70×80 + 2×8 = 5616.

HOW WE WORKED IT OUT: The units on the first two numbers have to add up to 10 to make this work and the second pair of numbers on the right hand side also have to add up to 10. The tens in the first product on the right hand side have to be 10 apart.

Suzanne Abbott & Nisha Doshi, (Y10) The Mount School, York proved that the method always works. Here is their solution:

Yes these are all correct. Here are some more examples

51×59 = 50×60 + 1×9 45×45 = 40×50 + 5×5

It always seems to work. Using algebra, we can try to generalise this

(10a + b)(10a + c) = 10a ×10(a + 1) + bc 100a2 +10ac +10ab + bc = 100a2 + 100a + bc 10ac + 10ab = 100a 10a(b + c) = 100a so b + c = 10

This fits with the examples we were given and the ones we have chosen.

This proof depends on the argument being reversible. Set out slightly differently it shows that the method works if and only if b + c = 10 and it is perfectly rigorous.

(10a + b)(10a + c) = 100a2 + 10ab + 10ac + bc = 100a2 + 100a + bc + 10ab + 10ac -100a = 100a2 + 100a + bc + 10a(b + c - 10) = 10a×10(a + 1) + bc