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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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?


The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

Christmas Chocolates

Age 11 to 14
Challenge Level

Why do this problem?

Many students are accustomed to using number patterns in order to generalise. This problem offers an alternative approach, challenging students to consider multiple ways of looking at the structure of the problem. The powerful insights from these multiple approaches can help us to derive general formulae, and can lead to students' appreciation of the equivalence of different algebraic expressions.

Possible approach

This problem might follow on nicely from Picturing Triangle Numbers, Mystic Rose, and Handshakes.

Display this image of a full size $5$ box of chocolates, and ask students to work out how many chocolates there are, without speaking or writing anything down. Compare solutions and share approaches.

Mention that mathematicians like to find efficient methods which can be used not only for simple cases but also when the numbers involved are very large. Explain that the pictures of Penny's, Tom's and Matthew's partially-eaten chocolates could be used by a mathematician as a starting point for finding an efficient method for counting the total number of chocolates.
Click here for the three pictures.

Hand out these chocolate box templates and ask students to show how the images of the partially-eaten chocolates can be used to calculate the total.

Ask students to report back, explaining the methods which have emerged.
Then ask students to use all three methods, along with any methods they devised for themselves, to work out the number of chocolates in a size $10$ box, and verify that all methods agree.

Challenge students to express each method for finding the number of chocolates in any size of box, perhaps introducing some algebra and the idea of a size $n$ box if appropriate.

Bring the class together to share findings. Compare the different "formulae" which have emerged, and ask students to explain why they are equivalent.

Key questions

How does each image help you to count the total number of chocolates quickly?
Can you demonstrate the equivalence of different algebraic expressions?

Possible extension

The problems Summing Squares and Picture Story lead to formulae for some intriguing sequences through analysis of the structure of the contexts.

Possible support

The problem Seven Squares gives lots of simple contexts where formulae emerge by looking at structure rather than number sequences.