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I'm Eight

Find a great variety of ways of asking questions which make 8.

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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.


Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Well done to Alex of Waingels Copse School, Reading, who cracked this Tough Nut. He ended up checking through every combination of the numbers and operations until he found the one that worked, using a spreadsheet to save a bit of time.

$$ \frac{8}{3 - 8\div3} = 24 $$

Before Alex found the solution above, lots of people tried bending the rules:

Callum (Madras College, St Andrew's, Scotland) and Bei(Riccarton High School, Christchurch, New Zealand) both noticed that the problem can be done with a square root sign:

$ \sqrt{8\times8}\times\sqrt{3\times3}$ or $\sqrt{(3\times8)\times(3\times8)} $

David (Alcester Grammar School) added in factorial notation:

$$ (3! \times 8) - (3 \times 8) $$

Ben(Madras College) used a bit of rotation:

$$ 8 \times 3 + \frac{3}{\infty} $$

Sarah(Madras College) decided that working in another base might help. Since there is an 8 in the problem, she decided to try base 9, and came up with:

$$ 38_{nine} - 3_{nine} - 8_{nine} = 26_{nine} = 24 $$

Tim Whitmore (Madras College) used recurring decimals:

$$ (8 + 8) \div (.\dot{3} + .\dot{3}) $$

Finally, Ravi, (St Xavier's College, Calcutta) used the greatest integer function, [x].
[x] is the greatest integer which is less than or equal to x.

$$ [(8\times8)\div3] + 3 = 24 $$