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#17 Semi-one numbers

Set by Colin Wright (with thanks to Adam Atkinson)

When you count up to 2, exactly half of the numbers contain the digit 鈥�1鈥�. We can use the term 鈥渟emi-one numbers鈥� to describe numbers with this property.

The same is true of a number between 15 and 20. Which one?

What is the next semi-one number after that?

Could there be infinitely many such numbers?

Solution next week

#16 Order, order!

Solution

There are many ways to arrange 1-9 so each pair differs by 2 or 3. For example: 1, 3, 5, 2, 4, 7, 9, 6, 8. For pairs to sum to a prime, we must alternate between odd and even numbers (so their sum is odd, since all primes are odd except for 2). We can find valid combinations with a graph in which numbers are connected by a line if they sum to a prime, and any valid route that visits all the numbers will work, such as 1, 2, 3, 4, 7, 6, 5, 8, 9. For adjacent pairs to form values in the times tables, we can determine possible locations for each digit 鈥� e.g. nothing in the times tables starts with a 9, so 9 must be at the end, and 6 can only be followed by 3 or 4. Combining these constraints will let us find the solution: 7, 2, 8, 1, 6, 3, 5, 4, 9.