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Solution to operator part of the course challenge

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What is the meaning of the house? That’s all that’s left to solve our challenge from the aliens and if we focus on what all of this is trying to tell us we really see that 7 in a house means - is equal to the number 142,857. And the number 7 in a house times 7 is equal to 999,999. Now, this must mean that the operator ‘house’ means the cyclic number corresponding to the number written inside it. So the meaning of the operator is the corresponding cyclic number, and what do we mean by corresponding cyclic number? Well first of all let’s just recap what a cyclic number is.
This is a cyclic number of length 6 and when we multiply it by the numbers 1,2,3,4,5 or 6, we just get back the same digits, shifted. For example, we saw that when we multiply this number by 4, we get the number 571,428. When we multiply it by 7 we get six nines - 999,999. There is one other property of cyclic numbers. And that is that they are the decimal expansion of one over the number, so, one seventh
is: 0.14285714285714… and so on and so on. It’s just the cyclic number repeating itself after the decimal point. This doesn’t happen with all the numbers, of course. One half is just 0.5. But it does happen with other numbers apart from 7. The next cyclic number up which we can write down in a house now is 17. It is a sixteen digit number, and it’s equal to, zero, putting in a leading zero here, that makes it simpler, 0588 - because when it’s shifted the zero get’s pushed in place - 0588235294117647 that is the 16-digit cyclic number corresponding to 17. And when you take this number and multiply it by 1-16 you’re going to get the same digits back, shifted.
So you can try that out for yourself on your computer. And one over seventeen is equal - you’ve already guessed it - to 0.0588235294117647 and of course it carries on 0588… and so on until eternity. So what are the aliens trying to tell us? That they know what cyclic numbers are? That they are highly intelligent! That they are very good at math! So we’re dealing here with an alien species that’s highly intelligent, knows math and loves recreational math - and that’s our challenge!
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Maths Puzzles: Cryptarithms, Symbologies and Secret Codes

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