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Summing Consecutive Numbers

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?

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The items in the shopping basket add and multiply to give the same amount. What could their prices be?


Can you work out how to win this game of Nim? Does it matter if you go first or second?


Age 11 to 14
Challenge Level

Why do this problem?

This problem challenges students to visualise what is going on as they figure out how the trick was done. Students' natural frustration in wanting to know how the trick is done may provide an opportunity for you as a teacher to let them struggle for longer than usual.

At the end of the task, once the vast majority of students have succeeded, you have an opportunity to celebrate their willingness to persevere and draw attention to the importance of resilience as a characteristic of good mathematicians.

Possible approach

Perform the trick for the class (or show the video). Hand out packs of cards so that each pair has one suit, and challenge students to work out how to order the cards to perform the trick.

Here is a worksheet with the three starting points from the problem.
Challenge students to figure out how each method works.
Finish the task by giving students a chance to explain and demonstrate each method.

Key questions

For Charlie's method: whereabouts must the Ace have been at the start of the trick?
For Luke's method: does it help to work backwards?
For Alison's method: why does the 3 come out first?

Possible extension

Use each method to work out the order needed to perform the trick in another language, or in reverse order King to Ace, or with two suits.

For more problems on visualising, see our Visualising collection.

Possible support

Charlie's method, together with paper and pencil for recording, is the most accessible.