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Summing Consecutive Numbers

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

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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?

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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?

Quick Times

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

$32 \times 38 = 30 \times 40 + 2 \times 8$

$34 \times 36 = 30 \times 40 + 4 \times 6$

$56 \times 54 = 50 \times 60 + 6 \times 4$

$73 \times77 = 70 \times80 + 3 \times 7$


And so on?

Verify and generalise if possible.