The wily explorer Knightmare, and his trusty companion, Slavy have scoured the deserts of the world in search of the fabled Icosahedron Diamond. Hundreds of miles from civilization, they discover the aforementioned bauble, a beautiful giant crystal worth a King’s ransom.

Unfortunately, the diamond is on a tiny island in the centre of a perfectly [...]

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Logic
What percentage of all integers contains at least one instance of the digit three?

For example, 13, 31, 33 and 103 all contain the digit “three” at least once.

Bonus points for your explanation…

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Maths
How quickly can you find out what is unusual about this paragraph? It looks so ordinary that you would think that nothing was wrong with it at all and, in fact, nothing is. But it is unusual. Why? If you study it and think about it you may find out, but I am not going [...]

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Tom
If you are walking southbound on a northbound bus, in how many directions are you traveling at that moment in time, ignoring micromovements? You must explain your answer.

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Higher Thinking,

Logic
You are lost in the woods. You come to a T-junction and are wondering which way to go. You have a funny feeling that one way will get you out, the other will mean certain death. Weird, eh?

Wizard of Oz approaches you. “A left turn will get you out of the woods,” he advises. [...]

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Probabilities
Whilst I was away under the pretense of work, a nearby bank uncovered a plot to swap the gold in their vaults with counterfeits. It was determined that all the gold bars in three of the Bank’s seven vaults were replaced with counterfeits. The other four vaults were uncompromised. The plot was foiled through the [...]

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Maths Logic
Prove that 2^n – 1 does not divide 3^n – 1 where n is any natural number > 1.

Solution: see post 54.

If m = 2 + 2 √(28 n² + 1) is an integer, where n is an integer, then show that m is a perfect square.

NB the convention is that the √ is positive.

I don’t know the solution.

I’d misread the previous one. I hope that I’ve got it right this time.

It should have been: solve in integers, the simultaneous equations:

x² = y² + y³ and x² + y² = z²

Solve in integers: x² = y² + y³

I don’t have an answer for this. But I have found some sweet intermediate results.