Tag Archives: elegance

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.

4s_2

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_aL_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.

61v1

\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.

61v2

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

Teaching Creativity in Mathematics

This will be the first of two ‘blog posts on an activity that could promote creativity for elementary, middle school, and high school students.  A suggestion for parents and teachers is in the middle of this post.

ABOUT A DECADE AGO, I first discovered what I call the Four 4s activity.  In brief, the game says that using exactly four 4s (no more, no less, and no other digits) and any mathematical operation you want, you can create every integer from 1 to 100.  Two quick simple examples are \displaystyle 3= \frac{4+4+4}{4} and \displaystyle 16= 4\cdot 4+4-4.

As for mathematical operations, anything goes!  The basic +, -, *, / along with exponents, roots, decimals (4.4 or .4), concatenation (44), percentages, repeating decimals (.\overline{4}), and many more are legal.

At the time, I was teaching a 7th grade prealgebra course with several students who were struggling to master order of operations–that pesky, but critical mathematical grammar topic that bedevils some students through high school and beyond.  I thought it would be a good way to motivate some of my students to 1) be creative, and 2) improve their order of operations abilities to find numbers others hadn’t found or to find unique approaches to some numbers.

My students learned that even within the strict rules of mathematical grammar, there is lots of room for creativity.  Sometimes (often? usually?) there are multiple ways of thinking about a problem, some clever and some blunt but effective.  People deserve respect and congratulations for clever, simple, and elegant solutions.  Seeing how others solve one problem (or number) can often grant insights into how to find other nearby solutions.  Perhaps most importantly, they learned to a small degree how to deal with frustration and to not give up just because an answer didn’t immediately reveal itself.  It took us a few weeks, but we eventually completed with great communal satisfaction our 1-100 integer list.

PARENTS and TEACHERS:  Try this game with your young ones or pursue it just for the fun of a mental challenge.  See what variations you can create.  Compare your solutions with your child, children, or student(s).  From my experiences, this activity has led many younger students to ask how repeating decimals, factorials, and other mathematical operations work.  After all, now there’s a clear purpose to learning, even if only for a “game.”

I’ve created an easy page for you to record your solutions.

A FEW WEEKS AGO, I read a recent post from the always great MathMunch about the IntegerMania site and its additional restriction on the activity–an exquisiteness scale.  My interpretation of “exquisiteness” is that a ‘premium’ is awarded to solutions that express an integer in the simplest, cleanest way possible.  Just like a simple, elegant explanation that gets to the heart of a problem is often considered “better”, the exquisiteness scale rewards simple, elegant formulations of integers over more complex forms.  The scale also includes surcharges for functions which presume the presence of other numbers not required to be explicitly written in common notation (like the 1, 2, & 3 in 4!, the 0 in front of .4, and the infinite 4s in .\overline{4}.

In the past, I simply asked students to create solutions of any kind.  I recorded their variations on a class Web site.  Over the past three weeks, I renamed exquisiteness to “complexity” and re-ran Four 4s across all of my high school junior and senior classes, always accepting new formulations of numbers that hadn’t been found yet, and (paralleling Integermania’s example) allowed a maximum 3 submissions per student per week to prevent a few super-active students from dominating the board.  Also following Integermania’s lead, I allowed any new submission to remain on the board for at least a week before it could be “sniped” by a “less complex” formulation.  I used differently colored index cards to indicate the base level of each submission.

Here are a few images of my students’ progress.  I opted for the physical bulletin board to force the game and advancements visible.  In the latter two images, you can see that, unlike Integermania, I layered later snipes of numbers so that the names of earlier submissions were still on the board, preserving the “first found” credit of the earliest formulations.  The boxed number in the upper left of each card is the complexity rating.

4s_1

4s_3

4s_2

The creativity output was strong, with contributions even from some who weren’t in my classes–friends of students curious about what their friends were so animatedly discussing.  Even my 3rd grade daughter offered some contributions, including a level 1.0 snipe, \displaystyle 5=\frac{4\cdot 4+4}{4} of a senior’s level 3.0 \displaystyle 5=4+\left( \frac{4}{4} \right)^4.  The 4th grade son of a colleague added several other formulations.

When obviously complicated solutions were posted early in a week, I heard several discussing ways to snipe in less complex solutions.  Occasionally, students would find an integer using only three 4s and had to find ways to cleverly dispose of the extra digit.  One of my sometimes struggling regular calculus students did this by adding 4′, the derivative of a constant. Another had already used a repeating decimal ( . \overline{4}), and realized she could just bury the extra 4 there ( .\overline{44}).  Two juniors dove into the complexity scale and learned more mathematics so they could deliberately create some of the most complicated solutions possible, even if just for a week before they were sniped.  Their ventures are the topic of my next post.

AFTERTHOUGHTS:  When I next use Four 4s with elementary or middle school students, I’m not sure I’d want to use the complexity scale.  I think getting lots of solutions visible and discussing the pros, cons, and insights of different approaches for those learning the grammar of mathematical operations would be far more valuable for that age.

The addition of the complexity scale definitely changed the game for my high school students.  Mine is a pretty academically competitive school, so most of the early energy went into finding snipes rather than new numbers.  I also liked how this game drove several conversations about mathematical elegance.

One conversation was particularly insightful.  My colleague’s 4th grade son proposed \displaystyle 1=\frac{44}{44} and argued that from his perspective, it was simpler than the level 1.0 \displaystyle \frac{4+4}{4+4} already on the board because his solution required two fewer operations.    From the complexity scale established at the start of the activity, his solution was a level 2.0 because it used concatenated 4s, but his larger point is definitely hard to refute and taught me that the next time I use this activity, I should engage my students in defining the complexity levels.

ADDENDA:

1) IntegerMania’s collection has extended the Four 4s list from 1 to well past 2000.  I wouldn’t have thought it possible to extend the streak so far, but the collection there shows a potential arrangement of Four 4s for every single integer from 1 to up to 1137 before breaking.  Impressive.  Click here to see the list, but don’t look quite yet if you want to explore for yourself.

As a colleague noted, it would be cool for those involved in the contest to see how their potential solutions stacked up against those submitted from around the world.  Can you create solutions to rival those already posted?

2) IntegerMania has several other ongoing and semi-retired competitions along the same lines including one using Four 1s, Four 9s, and another using Ramanujan’s ‘famous’ taxi cab number, 1729.  I’ve convinced some of my students to make contributions.

Play these yourself or with colleagues, students, and/or your children.  Above all, have fun, be creative, and learn something new.

It’s amazing what can be built from the simplest of assumptions.  That, after all, is what mathematics is all about.