# Unexpected Math Creativity Lessons

This is the second of two posts on my recent experiences with a Four 4s activity.  As I explained in my first post, I’ve used this activity for over a decade, but was re-inspired by a recent Math Munch post about an IntegerMania page playing a  Four 4s variation using Ramanujan’s 1729 taxi cab number.

What struck me was IntegerMania’s use of an exquisiteness level which I included in my recent Four 4s activity, calling it a “complexity scale” for my students.  I thought it a nice external measure of the difficulty of student constructions, but the scale drove several unexpected lessons.

Explaining Exquisiteness: Many students wanted to know why the mathematical functions and operations were leveled the way they were.  Hypothesizing the intent of the scale’s original author(s), I explained them as what one might expect to encounter as one’s mathematical understanding grew.

• Level 1.0 involves only single-digit 4s and the most basic math operations:   +,  -,  *,  and  /.
• Students bridge to Level 2.0 when they concatenate single digits (44 & 4.4) and use percentages.
• Level 3.0 introduces exponents and roots (which are really thinly-veiled exponents) and factorials.
• Level 4.0 opens high school math:  logarithms, trigonometry (circular and hyperbolic) and their inverses.

Mathematical Elegance:  I honestly thought my students would stop there.  While the formulation of the scale and “surcharges” (or ‘penalties’ as my students called them) were debatable and something I will work out as a group rather than imposing the next time I use this, they did reinforce some of what I’ve always discussed with my students.

• Any solution is better than no solution,
• Long or complicated solutions sometimes provide valuable insights and alternative perspectives on problems, and
• Once mathematicians begin to get a solid grasp on a situation, brief, elegant, often “minimalist” solutions that get directly to the core of an idea become the desired goal.

For these reasons, solutions with the lowest total “complexity” would be the solutions listed first on our collective Four 4s bulletin board.  My students called the replacement of any solution with a less complex solution sniping.  I thought their group goal would be to get solutions for all integers before sniping.  I was wrong.  They focused much more intently on sniping higher level solutions until we were down to fewer than 10 missing integers at which point there was a definite push to finish the list.  3-4 weeks after the activity started, our integer board is completed, and students continue to snipe existing solutions.

Unexpected Complexity:  Three of my students (juniors P and JP, & senior T) became absolutely entranced with some of the higher-level functions.  IntegerMania’s complete exquisiteness list contains more functions, but here are the ones these three primarily used, along with links to deeper explanations if needed.

• – They loved the Level 5.0 gamma function.  (For what it’s worth, I argue $\Gamma(4)=3!=6$ should be a higher level function because it ultimately relies on integral calculus, and IntegerMania lists derivatives as Level 6.0.)
– One even leveraged a matrix determinant to create a 61–a solution I pose below.
• Level 6.0 included
$p_a$ as the $a^{th}$ number in the list of prime numbers ($p_4=7$),
$f_a$ as the $a^{th}$ Fibonacci Number ($f_4=3$),
$\pi (a)$, the Prime Counting Function which conveniently is a Wolfram Alpha function,
$d(a)$, the number of divisors of a,
$\sigma(a)$, the sum of the divisors of a,
Euler’s totient function, $\phi (a)$, “the number of positive integers less than or equal to a that are relatively prime to a“–also a Wolfram Alpha function, and
– the derivative from calculus, allowing a convenient way to lose an extra 4 because 4’=0.
• Finally, some Level 7.0 favored functions:
Double factorials with $4!!=4\cdot 2=8$,
– the Lucas Numbers, $L_a$$L_4=7$, and
– the Triangular Numbers, $T_a$, a sort of stealthy use of combinations where $T_4=10$.

Strategizing:  A couple days into the activity, P and JP set themselves a goal of writing every integer from 0-25 with a single 4.  Enamored with the possibility of using their newfound functions, they realized that if they could accomplish this goal, they could write every integer 1-100 on the board with four 4s.  It didn’t matter to them that the complexity levels would be high, they wanted to prove to themselves that every answer could be found without actually finding each–in short, they sought a form of an existence proof long before all answers were posted.  I didn’t anticipate this, but loved their approach.

Here’s a reproduction of their list:

• $0 = 4'$
• $1 = \Gamma \left( \sqrt{4} \right)$
• $2 = \sqrt{4}$
• $3 = f_4$, T made huge use of this one.
• $4 = 4$
• $5 = p_3=p_{d(4)}$
• $6 = \Gamma(4)$
• $7 = L_4$
• $8 = 4!!$
• $9 = \pi(24)=\pi(4!)$
• $10 = T_4$
• $\displaystyle 11 = L_5 = L_{p_{d(4)}}$
• $12 = \sigma(6) = \sigma \left( \Gamma(4) \right)$
• $13 = \sigma(9) = \sigma \left( \pi (4!) \right)$
• $14 = \pi(45) = \pi \left( T_9 \right) = \pi \left( T_{ \pi (4!) } \right)$
• $15 = \sigma(8) = \sigma(4!!)$
• $16 = \pi(55) = \pi(f_{10}) = \pi \left( f_{T_4} \right)$
• $17 = p_7 = p_{f_4}$
• $18 = \sigma(10) = \sigma(T_4)$
• $19 = p_8 = p_{ (4!!) }$
• $20 = \phi(25) = \phi(\pi(\sigma(\phi(\phi(p_{(T_{(f_4)})})))))$
• $21 = f_8 = f_{4!!}$
• $22 = \phi(23) = \phi( p_{ \pi(4!!) } )$
• $23 = p_9 = p_{ \pi(4!) }$
• $24 = 4!$
• $25 = \pi(98) = \pi(\sigma(52)) = \pi(\sigma(\phi(106))) = \pi(\sigma(\phi(\phi(107))))$
$= \pi(\sigma(\phi(\phi(p_{28})))) = \pi(\sigma(\phi(\phi(p_{(T_7)})))) = \pi(\sigma(\phi(\phi(p_{(T_{(f_4)})}))))$

That 25 formulation is a beast (as is the 20 that depends on it), but P and JP accomplished their goal and had proven that the entire board was possible.

Now, all that remained for the class was to find less complex versions.

A Creative Version of 61:  As my sign-off, I thought you might enjoy JP’s use of a determinant and some Level 6.0 functions to create his 61.  He told me he knew it would be sniped, but that wasn’t the point.  He just wanted to use a determinant.

$\pi (4!)=\pi (24)=9$ because there are 9 primes less than or equal to 24, and $p_9=23$ because the 9th prime number is 23.  With $f_4=3$ from above, the remainder of the determinant is easily handled.  The prime number functions were a base Level 6.0, and the surcharges for each of them, the factorial, the implied 2 on the root, and the Fibonacci function raised this to a Level 7.0.

A little over a week later, JP’s determinant was sniped by a student who isn’t even in my classes, N, whose Level 3.4 construction follows.

I hope you can have some fun with this, too.