# Just a Simple Number Crossword

I made a small number crossword for my mathletes club that I thought I'd share. It's not too hard, I think it's just right for a coffee break. Have fun! • Is the last across clue intended to be enigmatic? And isn't 4A true of every integer? – noedne Apr 25 '19 at 21:32
• @noedne How is (e.g.) 3 divisible by two distinct perfect cubes? – GentlePurpleRain Apr 25 '19 at 21:42
• @GentlePurpleRain $1^3$ and $(-1)^3$ – noedne Apr 25 '19 at 21:43
• I'm pretty sure that when discussing divisibility, it is generally assumed that one is dealing only with positive integers. – GentlePurpleRain Apr 25 '19 at 21:45
• @noedne yes, I only want positive factors. – greenturtle3141 Apr 25 '19 at 22:06

$$\begin{array}{|c|c|c|} \hline 7 & 1 & 8 \\ \hline 1 & 3 & 5 \\ \hline \end{array}$$

My thinking went something like this:

• 3D means that the second digit of 1D must be a $$1$$, because any number 20 or larger multiplied by 5 would end up as 3 digits. (It can't be a 0 because leading 0s aren't allowed for 4A.)
• Since 1 is reversible on a 7-segment display, the other digit of 1D must not be. The only digits that don't form a digit when turned upside down are 3, 4, and 7. Thus 3D must be $$65$$, $$70$$, or $$85$$.
• For 4A, we now have a first digit of $$1$$, and a last digit of $$5$$ or $$0$$. The middle digit must be odd, because 2D is prime, and no 2-digit prime is even. This makes the number of possibilities for 4A very small. By a little trial and error (multiplying 8, 27, 64), it easy to discover that the only possible solution is $$135$$, which is a multiple of both $$1^3$$ and $$3^3$$.
• Now we have that 1A is either $$3x6$$ or $$7x8$$ (where $$x$$ is the unknown digit). Since two of the digits must sum to the third, the only possibilities are $$336$$, $$396$$, or $$718$$. The first two options do not make 2D prime (33 and 93 are both composite), so the solution for 1A must be $$718$$, making 2D $$13$$.

• The final answer (thanks to PiIsNot3, below), is found by

using A1Z26 to translate each number into a letter. The dotted line indicates that those two digits should be treated as a single number.
Thus we have 7-18-1-3-5, which translates to

GRACE

• This is the correct numerical answer. – greenturtle3141 Apr 25 '19 at 22:06
• Notice that the line separating 2 and 3 is dotted, meaning that the 1 and 8 should be read together as 18. Doing A1Z26 gives rot13(TENPR), which is the “elegant name.” – HTM Apr 25 '19 at 22:08
• I have added an explanation and the final answer, as supplied by @PiIsNot3. – GentlePurpleRain Apr 26 '19 at 15:09

Speaking strictly, there is another (numerical) solution:

145
135

That's because

On a 7-segment indicator, an upside-down 1 is not the same that the "original" 1.

• There is no such requirement in the puzzle. – jarnbjo Apr 26 '19 at 13:16
• You are technically right, that clue may be somewhat vague. In spirit, what I wanted to get at was the fact that numbers such as 219, 6901, 55, etc. can be cleanly read as numbers when turned upside-down, especially when entered into a 7-segment display. However, I will argue here that even an "11" would still technically represent a base 10 numeral, even though the placement of the ones don't follow the standard convention. The same cannot be argued for numbers such as 3, 4, or 7, and assigning values to "E", "h", or "L" seems a bit of a stretch, so I think we're in the clear here. – greenturtle3141 Apr 26 '19 at 16:05