BY THE NUMBERS
DAVID RICHESON

Here's a crossnumber puzzle by Professor David Richeson, Professor of Mathematics and the John J. Curley ’60 and Ann Conser Curley ’63 Faculty Chair in the Liberal Arts at Dickinson College. It's one of the best crossnumber puzzles I've seen besides my own out there, and incorporates a lot of mathematical topics that are a little out of the mainstream. I've recently solved the puzzle, and am going to comment on some of the clues below to show how solving a puzzle such as this can be an enlightening learning experience for all kinds of students and the general public.

I should say here that the styles for this page come from the Mathpix Snip app, which has excellent math and science fonts and styles. I recommend it for getting math and science content onto the web.

The apps I used to solve the puzzle were: Perplexity AI (the free version), Wolfram Alpha (the free version) and Microsoft Math Solver (iOS version). I also used a regular search engine on the web for some of these.

Here are the instructions and the blank grid and the clues of this puzzle. I will comment on several of the clues directly in the clue text.

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Fill in the entries of this crossword puzzle with a decimal point "." or a digit 0 through 9. All decimal values that are longer than the given number of blanks are truncated, not rounded.

ACROSS

1 ${15}^6+{189}^2$${15}^6+{189}^2$15^(6)+189^(2)15^{6}+189^{2}${15}^6+{189}^2$
6 Sum and product of its digits are equal
10 Number of primes less than 10,000
14 Acceleration of gravity in $\mathrm{m}/{\mathrm{s}}^2$$\mathrm{m}/{\mathrm{s}}^2$m//s^(2)\mathrm{m} / \mathrm{s}^{2}$\mathrm{m}/{\mathrm{s}}^2$
15 $\ln (2)/{\pi }^2$$\ln (2)/{\pi }^2$ln(2)//pi^(2)\ln (2) / \pi^{2}$\ln (2)/{\pi }^2$
16 Speed of light ____ $\times {10}^8\mathrm{m}/\mathrm{s}$$\times {10}^8\mathrm{m}/\mathrm{s}$xx10^(8)m//s\times 10^{8} \mathrm{~m} / \mathrm{s}$\times {10}^8\mathrm{m}/\mathrm{s}$
17 $(12+\sqrt{130})\pi$$(12+\sqrt{130})\pi$(12+sqrt130)pi(12+\sqrt{130}) \pi$(12+\sqrt{130})\pi$
18 Number of rook moves on a $16\times 16$$16\times 16$16 xx1616 \times 16$16\times 16$ board
[PerplexityAI got me the answer to this one.]
19 Hardy's cab number
[This is a reference to a story about Ramanujan and his taxicab ride to the hospital. Again, PerplexityAI to the rescue!]
20 $0,1,4,\dots$$0,1,4,\dots$0,1,4,dots0,1,4, \ldots$0,1,4,\dots$
23 Water freezing temperature in Kelvin
24 A palindromic permutable prime
[Palindromic indicates the same backwards and forwards, and a permutable prime number is defined here by PerplexityAI.]
25 $2+\frac{1}{1+\frac{1}{56+\frac{1}{7}}}$$2+\frac{1}{1+\frac{1}{56+\frac{1}{7}}}$2+(1)/(1+(1)/(56+(1)/(7)))2+\frac{1}{1+\frac{1}{56+\frac{1}{7}}}$2+\frac{1}{1+\frac{1}{56+\frac{1}{7}}}$
28 Sum of the first four 4th powers
29 Young Gauss's sum
31 Diagonal of a square with the same area as the unit circle
[I really like geometry problems as clues for crossnumber puzzles! The free version of PerplexityAI doesn't generate images in the result, but the links in the answer to my query show the images.]
32 Look-and-say sequence
36 $1000(30+{\pi }^2)$$1000\left(30+{\pi }^2\right)$1000(30+pi^(2))1000\left(30+\pi^{2}\right)$1000(30+{\pi }^2)$
38 $10$$10$1010$10$ times the Basel problem sum
39 $2,3,5,$$2,3,5,$2,3,5,2,3,5,$2,3,5,$ ___
41 Smallest number whose square has eight digits
42 Location of the absolute minimum of $g(x)=x^6-420x+100$$g(x)=x^6-420x+100$g(x)=x^(6)-420 x+100g(x)=x^{6}-420 x+100$g(x)=x^6-420x+100$
44 $\tau$$\tau$tau\tau$\tau$
46 Emergency number in the US
47 $100e^e$$100e^e$100e^(e)100 e^{e}$100e^e$
49 Bronze ratio: $1+\sqrt{3+\sqrt{3+\sqrt{3+\dots }}}$$1+\sqrt{3+\sqrt{3+\sqrt{3+\dots }}}$1+sqrt(3+sqrt(3+sqrt(3+dots)))1+\sqrt{3+\sqrt{3+\sqrt{3+\ldots}}}$1+\sqrt{3+\sqrt{3+\sqrt{3+\dots }}}$
[The bronze ratio can be represented as above, or as a continued fraction (similar to the clue for 25 Across, but going on forever), and a few other ways. There's a really good explanation of all the metallic ratios on the Rosetta Code website. This page gives the ratios elemental metal names past gold, silver and bronze, all the way to lead! The Wikipedia page on metallic means is also a good place to start for understanding the concept. Cool stuff!]
50 In Egyptian hieroglyphics:

$\cap \cap \cap \cap \cap \cap \cap \cap \cap \parallel$$\cap \cap \cap \cap \cap \cap \cap \cap \cap \parallel$nn nn nn nn nn nn nn nn nn||\cap \cap \cap \cap \cap \cap \cap \cap \cap \|$\cap \cap \cap \cap \cap \cap \cap \cap \cap \parallel$
52 Every $3x+1$$3x+1$3x+13 x+1$3x+1$ sequence ends this way?
53 Fourth Fermat prime
56 $\pi$$\pi$pi\pi$\pi$
60 The 24 th Mersenne prime has this many digits
62 10th Pell number
63 ${\pi }^e$${\pi }^e$pi^(e)\pi^{e}${\pi }^e$
64 $30°-$$30°-$30°-30°-$30°-$ ___$°-$$°-$°-°-$°-$ ___$°$$°$°°$°$ right triangle
65 Has digit sum 20 and digit product 336
66 $a,b,c$$a,b,c$a,b,ca, b, c$a,b,c$ where $\frac{1449}{50}=a+\frac{1}{b+\frac{1}{c}}$$\frac{1449}{50}=a+\frac{1}{b+\frac{1}{c}}$(1449)/(50)=a+(1)/(b+(1)/(c))\frac{1449}{50}=a+\frac{1}{b+\frac{1}{c}}$\frac{1449}{50}=a+\frac{1}{b+\frac{1}{c}}$
67 In Mayan numerals:
$\stackrel{⋮}{\equiv }$$\stackrel{⋮}{\equiv }$-=^(vdots)\stackrel{\vdots}{\equiv}$\stackrel{⋮}{\equiv }$

68 Fourth perfect number
69 Hypotenuse of a Pythagorean triple with sides 27,560 and 64,791

DOWN

2 A fixed point of $f(x)=2x^{10}-23x$$f(x)=2x^{10}-23x$f(x)=2x^(10)-23 xf(x)=2 x^{10}-23 x$f(x)=2x^{10}-23x$
3 $3825/99=$$3825/99=$3825//99=3825 / 99=$3825/99=$ ___ . ___
4 $\lceil \sinh (9)\rceil$$\lceil \sinh (9)\rceil$|~sinh(9)~|\lceil\sinh (9)\rceil$\lceil \sinh (9)\rceil$
5 $11\cdot {10}^3(600+\sqrt[3]{e})$$11\cdot {10}^3(600+\sqrt[3]{e})$11*10^(3)(600+root(3)(e))11 \cdot 10^{3}(600+\sqrt[3]{e})$11\cdot {10}^3(600+\sqrt[3]{e})$
6 The Sophie Germain prime that generates the safe prime 81,527
7 An eigenvalue for $\left(\begin{array}{cc}2& 1\\ 3& -2\end{array}\right)$$\left(\begin{array}{cc}2& 1\\ 3& -2\end{array}\right)$([2,1],[3,-2])\left(\begin{array}{cc}2 & 1 \\ 3 & -2\end{array}\right)$\left(\begin{array}{cc}2& 1\\ 3& -2\end{array}\right)$
8 It is $1/9$$1/9$1//91 / 9$1/9$th its reverse
9 $45$$45$4545$45$th term in Padovan sequence, which starts $1,1,1$$1,1,1$1,1,11,1,1$1,1,1$ and $p_n=p_{n-2}+p_{n-3}$$p_n=p_{n-2}+p_{n-3}$p_(n)=p_(n-2)+p_(n-3)p_{n}=p_{n-2}+p_{n-3}$p_n=p_{n-2}+p_{n-3}$
10 XMMCLXXXIX
11 $e$$e$ee$e$
12 Number of ways to make change for a dollar without the $\mathrm{\$}1$$\mathrm{\$}1$$1\$ 1$\mathrm{\$}1$ coin
13 Emergency number in the UK
21 In Morse code:

22 Circumference of a circle of radius $9/14$$9/14$9//149 / 14$9/14$
26 Gelfond-Schneider constant: $2^{\sqrt{2}}$$2^{\sqrt{2}}$2^(sqrt2)2^{\sqrt{2}}$2^{\sqrt{2}}$
27 Zip code for Elkton, Minnesota (population 141)
28 It is CAB in hexadecimal
29 Slope of the tangent line to $y=\frac{1}{10}{x}^{10}-\ln x$$y=\frac{1}{10}{x}^{10}-\ln x$y=(1)/(10)x^(10)-ln xy=\frac{1}{10} x^{10}-\ln x$y=\frac{1}{10}{x}^{10}-\ln x$ at $x=2$$x=2$x=2x=2$x=2$
30 Fibonacci sequence starting with no rabbits
32 $1366$$1366$13661366$1366$th prime number
33 Diagonal of a regular pentagon with side-length $10$$10$1010$10$
34 $1+1/\pi$$1+1/\pi$1+1//pi1+1 / \pi$1+1/\pi$
35 A prime factor of $\mathrm{111,111,111,111,111,111,111,111,111,111}$$\mathrm{111,111,111,111,111,111,111,111,111,111}$111,111,111,111,111,111,111,111,111,111111{,}111{,}111{,}111{,}111{,}111{,}111{,}111{,}111{,}111$\mathrm{111,111,111,111,111,111,111,111,111,111}$
[Here's an example where Wolfram Alpha comes in handy. If you just input a number into Wolfram Alpha, it outputs a range of properties of the number, including its prime factorization. See here for this number.]
37 Append any of its digits to the end and it is prime
40 In Chinese counting rods:

43 Fifth Mersenne prime
45 ${\int }_0^{10}\frac{1}{2\sqrt{x}}dx$${\int }_0^{10} \frac{1}{2\sqrt{x}}dx$int_(0)^(10)(1)/(2sqrtx)dx\int_{0}^{10} \frac{1}{2 \sqrt{x}} d x${\int }_0^{10}\frac{1}{2\sqrt{x}}dx$
48 Number of sequences of 20 coin tosses starting with heads
51 $i^i$$i^i$i^(i)i^{i}$i^i$
[This is my favorite clue in this puzzle! It turns out that $i^i$$i^i$i^(i)i^{i}$i^i$ is a real number! PerplexityAI explains it like this.]
52 In Babylonian cuneiform:
53 $e^{\pi }$$e^{\pi }$e^(pi)e^{\pi}$e^{\pi }$
54 Number of 5-letter codes made from A through K with no repetition
55 It is $10011001100000111$$10011001100000111$1001100110000011110011001100000111$10011001100000111$ in binary
57 Fourth row of Pascal's triangle
58 Euler-Mascheroni constant: $\gamma =0$$\gamma =0$gamma=0\gamma=0$\gamma =0$. ___.
59 Feet in a mile
60 Devil's number
61 James Bond's number