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These secondary problems have not yet been solved. Can you be the first?

Three Consecutive Odd Numbers

How many sets of three consecutive odd numbers can you find, in which all three numbers are prime?

Adding Odd Numbers

Is there a quick and easy way to calculate the sum of the first 100 odd numbers?

Different Products

Age 14 to 16
Challenge Level

 

Take any set of four consecutive whole numbers.
Find the product of the first and last numbers.
Find the product of the second and penultimate (in this case third!) numbers.

e.g. consider the set $5, 6, 7, 8$
       $5 \times 8 = 40$
       $6 \times 7 = 42$

Try a few more examples with sets of four consecutive numbers.
Do your results have anything in common?

Now take a set of five consecutive whole numbers.
Again, find the product of the first and last numbers, and the product of the second and penultimate numbers.
Try several examples.
Do your results have anything in common?

Can you predict the difference between the product of the first and last numbers, and the product of the second and penultimate numbers, when you have 6 consecutive whole numbers? 
What if you have $7$, or $10$, or $20$, or $100$ consecutive whole numbers?

Select sets of consecutive numbers and test your predictions.

Can you predict what the difference will be when you have $n$ consecutive whole numbers?

 

Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true, and look to proofs to provide rigorous and convincing arguments and justifications.

Can you prove that when you have n consecutive numbers, the difference between the product of the first and last numbers, and the product of the second and penultimate numbers, will be $n-2$?

Below is a proof that has been scrambled up.
Can you rearrange it into its original order?

 

If you can find a proof which is different to the one in our proof sorter, then please do let us know by submitting it as a solution.

Extension:

Can you prove that when you have $n$ consecutive even numbers, the difference between the product of the first and last numbers, and the product of the second and penultimate numbers, will be $4n-8$?

 

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.