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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?

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?

Fibs

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?

Mindreader

Age 11 to 14 Challenge Level:

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers with none greater than 60 (say for example she picks 42, 43, 44). Ask her to tell you a multiple of 3 that is less than 100 (for example 39). Then ask her to add the four numbers and multiply by 67, not letting you see the numbers of course, and to tell you the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.

This is what you do. Your friend gives you a 2 digit number and you double it to get the remaining digits. In this case the calculation is

42+ 43 + 44 + 39 = 168, 168 x 67 = 11256

You are told the last two digits, namely 56, and you can immediately give the whole answer 11256.

To get the three consecutive numbers you divide her multiple of 3 by 3 and add 1 (e.g. 13+1=14). Subtract this from the two digit number she has given you and you get the first of the three consecutive numbers in question (e.g. 56 -14 = 42).

Explain why this works.