Tag Archives: creativity

Recentering Normal Curves, revisited

I wrote here about using a CAS to determine a the new mean of a recentered normal curve from an AP Statistics exam question from the last decade.  My initial post shared my ideas on using CAS technology to determine the new center.  After hearing some of my students’ attempts to solve the problem, I believe they took a simpler, more intuitive approach than I had proposed.


In the first part of the problem, solvers found the mean and standard deviation of the wait time of one train: \mu = 30 and \sigma = \sqrt{500}, respectively.  Then, students computed the probability of waiting to be 0.910144.


The final part of the question asked how long that train would have to be delayed to make that wait time 0.01.  Here’s where my solution diverged from my students’ approach.  Being comfortable with transformations, I thought of the solution as the original time less some unknown delay which was easily solved on our CAS.



Instead of thinking of the delay–the explicit goal of the AP question–my students  sought the new starting time.  Now that I’ve thought more about it, knowing the new time when the train will leave does seem like a more natural question and avoids the more awkward expression I used for the center.

The setup is the same, but now the new unknown variable, the center of the translated normal curve, is newtime.  Using their CAS solve command, they found


It was a little different to think about negative time, but they found the difference between the new time difference (-52.0187 minutes) and the original (30 minutes) to be 82.0187 minutes, the same solution I discovered using transformations.


This is nothing revolutionary, but my students’ thought processes were cleaner than mine.  And fresh thinking is always worth celebrating.

CAS Presentations at USACAS-9

I had two presentations at last Saturday’s USACAS-9 conference at Hawken School in Cleveland, OH.  Following are outline descriptions of the two sessions with links to the PowerPoint, pdf, and .tns files I used.  I’m also adding all of this information to the Conference Presentations tab of this ‘blog.

Powerful Student Proofs

This session started with a brief introduction to a lab that first caught my eye at the first USACAS conference years ago.

You know how the graph of y=ax^2+bx+c behaves when you vary a and c, but what happens when you change b?

I ‘blogged on this problem here and here.  In the session, we used TI-Nspire file QuadExplore.

Next, we explored briefly the same review of trigonometric and polar graphs not as static parent functions under static transformations, but as dynamic curves oscillating between their ceilings and floors.  In the session, we used TI-Nspire file Intro Polar.

Having a complete grasp of polar graphs of limacons, cardioids, rose curves, and hybrids of these, I investigated what would happen for curves of the family r=cos \left( \frac{\theta}{k} \right).  Curiously, for k=3, I encountered a curve that looked like a horizontal translation of limacons–something that just shouldn’t happen within polar coordinates.


One of my former students, Sara, used a CAS to convert a polar curve to Cartesian, translate the curve, and convert back to polar.  She then identified and solved a trig identity to confirm what the graph suggested.  A complete description of Sara’s proof is below.  I originally ‘blogged on Sara’s work here which was a much more elegant solution to the problem than my initial attempt.  It’s always cool when a student’s work is better than her teacher’s!  I used TI-Nspire file Polar Fractions in Saturday’s session.

The last example presented itself when I created a document to model the family of conic curves resulting from manipulating the coefficients of Ax^2+Bxy+Cy^2+Dx+Ey+F=0.  After I created  dynamic points for the foci of the conics, something unusual happened when the E parameter for horizontal ellipses and hyperbolas varied.



The foci for hyperbolas followed an ellipse, and the locus of elliptical foci appeared to be a hyperbola.  Another former student, Lilly, proved this property to be true.  A detailed explanation of Lilly’s proof is below.  We were fortunate to have Lilly’s work published in the Mathematics Teacher in May, 2014.

To demonstrate this final part of the session, I used TI-Nspire file Hidden Conic Behavior.

Here is my PowerPoint file for Powerful Student Proofs.  A more detailed sketch of the session and the student proofs is below.

Bending Asymptotes, Bouncing Off Infinity, and Going Beyond

The basic proposal was that adding the Reciprocal transformation to the palette of constant dilations and translations dramatically simplified understanding of the behavior of rational functions around even and odd vertical asymptotes (bouncing off and passing through infinity).  Just like lead coefficients of polynomials determine their end behavior, so, too, do the lead coefficients of proper rational expressions define the end behavior of rational functions.

Extending the idea of reciprocating and transforming functions, you can quickly explain exponential decay from exponential growth, derive the graphs of y=\frac{1}{x} and y=\frac{1}{x^2}, and completely explain why logistic functions behave the way they do.

We finished with a quick exploration of trigonometric and polar graphs not as static parent functions under static transformations, but as dynamic curves oscillating between their ceilings and floors.

I used TI-Nspire Bending and Intro Polar files in the demonstration.  Here is my outline PowerPoint file for Bending Asymptotes.

“Math Play” Presentation for Early Childhood teachers

Even though my teaching experiences are all middle and high school, as a PreSchool-12 math chair and father of 3 young children, I’m intensely interested in how math is presented to very young people.

As a result, I’m presenting ideas for teaching math through fun and exploration to about 55 Cleveland area pre-school through kindergarten teachers this morning.  My handout is on Scribd and should show below.  Math is  about Play and Curiosity.  Teach it that way.


Optimization in Four Colors

I suspect many (most?) geometry teachers know of the Four Color Theorem (FCT) which roughly states that any flat map, no matter how complex, containing only contiguous regions with finite perimeter can be colored with no more than four colors with the only restriction for coloring being that regions have different colors if they share any boundary beyond a set of finite points. While the FCT is not a particularly useful to cartographers, it has historical significance as the first significant mathematical proof to have been established with extensive use of computers.

THE PROBLEM:  In secondary geometry classes, the FCT is typically just a footnote or factoid, but it is pretty easy to understand for students of all levels.  This year I decided to make it more interesting as an optimization problem.  If each color you use has a different “cost” per area unit, can you color a given map as “cheaply” as possible?

[I considered a maximum cost map, too, and convinced myself that the maximum cost map is just a flip of all the colors, assuming the change in cost is the same between all colors.  With that thought, saving money seemed the more “realistic” goal, so I went with minimum cost.]

MOTIVATIONS:  Perhaps the BEST part of this project was that I was not–and still am not–convinced that we have found THE optimal solution.  I was reasonably certain that I could determine a very good mapping cost, but the sheer number of possibilities would require significant computer run time and coding abilities (just like the original FCT proof!) to ferret out the best answer–resources not available to those solving the problem (the computing problem is an issue my school is actively addressing).  I loved having a problem in math where determined students might best their teacher–and some did!!  I also liked that this project significantly motivated my students to use spreadsheets to track their data–a different math resource than most were accustomed to using.

IMPLEMENTATION:  Experimenting, I decided to offer different versions of the coloring challenge to my 4th-5th grade math club and all of my 8th grade math classes (prealgebra, algebra, & geometry).

Project 1:  Our 8th grade humanities course had an Africa unit earlier in the year, so I returned there by asking all of the students to color this map of Africa.


We provided this spreadsheet of country names and areas along with these coloring costs:  Purple = $2.00/mi^2, Yellow=$2.50/mi^2, Red = $3.00/mi^2, and Blue=$3.50/mi^2.  After some discussions on the first day, the “border” rule was revised to note that countries whose borders were only large lakes (Democratic Republic of the Congo & Tanzania plus Chad & Nigeria) could be considered “not touching” for this project.

Political incorrectness confession:  We noticed a day after we assigned the project that we had inadvertently left off the relatively new South Sudan. I decided to leave the two Sudans as a single country for this exercise (thus the inked in portion of the map).  Having compromised the previous day on the lake-bordered countries, my error accidentally made the largest and 3rd largest African countries border each other–a nice confounding problem, I thought, for forcing students to determine which would get a cheaper color.

Project 2:  I gave the relatively simpler map of the lower 48 US States to our 4th-5th grade math club with coloring restrictions Red=$1.00, Yellow=$1.25, Blue=$1.50, and Green=$1.75.


RESULTS:  For the submission, students (in ones or pairs) had to submit their colored map, a spreadsheet showing their computations, and 1-3 paragraphs explaining their general coloring strategies, and especially how they handled the inevitable situations where their coloring strategies self-conflicted.  In general, we could have done a better job preparing students for the written portion, but the two most commonly stated strategies were

1. (Low level) We colored the biggest countries the cheapest as far as we could, and then colored the next largest using the next cheapest color, etc.  If we ran into conflicts, we “worked it out”.

2. (Stronger) Noticed after trying the obvious strategy above that the countries colored the 2nd cheapest surrounding a “cheapest color” country often had more area than the “cheapest color” country.  By paying a little more for the largest country, they more than made up for the added expense by coloring a collection of countries that in total had more area.

3. A  few members of my math club addressed some specific strategies like the 11-state ring of US states (MO-IL-IN-OH-WV-VA-NC-GA-AL-MS-AR) surrounding Kentucky & Tennessee made it possible to use just two alternating colors over a large area.

Using our color schemes, excellent scores for the US map were very close to, but just over $1,000,000.  The best Africa map scores we found were just under $17,000,000.  As I noted earlier, I’m not at all convinced that we have found the optimum values, but part of the fun of these projects was that anyone with some calm logic and determination could break through.  My second best coloring scheme was from a student who had been exposed to the least amount of math.  If you can beat these scores, please share.

VARIATIONS:  After playing with this for a while, I’m convinced that all optimal solutions depend on the gap you set between the color costs.  The more expensive the next color is, the more motivation you have to not change colors.  I haven’t tried it, but I think strategy #2 above could be exploited more often if paint color jumps are smaller on a large, complicated map.

I’m also convinced that the initial paint cost is irrelevant.  It will change the total cost of the project, but it would just scale all values up or down.

I didn’t play with different step values in paint cost, but I can see that potentially changing the game, especially if the cost jumps increase as you approach the 4th color.


Where is 74?

From some earlier posts (here, here, and here), you know I’m rather fond of the Four 4s puzzle.  I’ve invoked it twice in the first month of this school year.

Following are descriptions of how I’m using it first with middle school students  and then a twist I employed for some interested 4th & 5th graders.

Stage 1:  8th Grade

I began the Four 4s puzzle with my 8th graders partly as a way to reinforce order of operations after a summer off from math (for most of the students), but primarily to bring home the ideas that mathematics absolutely has room for creativity, and that there are often multiple ways to solve problems.  Last week, the final integers from 0-100 were found by Hawken’s 8th graders:


One of the great by-products of this game is the discussion of what solutions would be considered “better” or “simpler” than others.  Based on suggestions from my students last year, this time around I asked this year’s participants to set their own levels for the math operations. Expectedly, the basic four operations and absolute values were Level 1.  Concatenation (44) and decimals (.4) were deemed Level 2, as were exponents.  I was initially surprised when the students then declared square roots to be Level 3.  Even though I consider them equivalent in complexity to exponents, this group of 8th graders consider the root operation more difficult–a good lesson for me this year.

A few remembered factorials (4!), called them Level 3, and actively used them to access larger numbers.  When the last few numbers were proving difficult, some began researching other math operations.  One discovered the Gamma Function, but the cool surprise was the introduction of primorials–prime factorials–a function I’d never explored before.  There are two primorial definitions; the students chose the one where you multiply all prime numbers less than or equal to the number being “primorialed”.  For example, 4\# =3\cdot 2=6 and (4\# ) \#=6 \# =5\cdot 3\cdot 2 = 30.  (4\# )\# was used pretty heavily by one student.

As a final touch, I used red paper to indicate Level 1 answers and gray paper for Level 2 as a way to visually highlight the lowest level, ideal answers.  (My new school’s colors are red and gray.)  I was also pleased with the way many students paid attention to how some numbers were solved and leveraged those techniques to solve or improve other numbers.  As the numbers are all now located, some are still trying to lower the levels–sniping.  I’m still uploading the list of their lowest level findings, but I’m eventually going to have all of the “simplest versions” of the 8th grade Four 4s findings here.

Stage 2:  4th & 5th Grade

I was asked to start an after-school “Math Club” this year for interested students in our 4th & 5th grades.  Thinking it would be a cool way to stretch some younger students, I decided to play a variation of Four 4s.  Hawken was founded in 1915, so inspired by Integermania‘s variations, we made a new game with the same rules and different digits:

Using only the digits 1, 9, 1, & 5,
make every integer from 0-100.

1915_1In homage to Hawken’s impending 100th anniversary, we gave special recognition to any solution that used the four digits in 1915 order.

Before starting, I checked the problem’s viability by generating about 90 of the integers from 0-100 before deliberately stopping to be able to tell the students honestly that they had an opportunity to find some numbers I hadn’t–a cool pride point for some to be able to find what their teachers couldn’t.  I also decided not to introduce sniping for this group because I wanted to encourage more creativity and cooperation.

Here is the list of what our students have discovered.  The students have used decimals, two-digit numbers, and square roots, and have even learned factorials.  Monday, I introduced repeated digits to help them go a bit further.

When 18 numbers remained unfound last week, the group decided some of their solutions on a big wall where everyone in the lower school would pass to generate interest in math creativity.  They chose not to post everything they had found in hopes that others would be able to find solutions the group hadn’t discovered.

1915_3The numbers written in red are the more valuable “1915” solutions; the gray are other valid solutions.  We also decided that it would be cool to create a “most wanted” set of posters (think Old West) for the numbers who were still “on the loose”.


Within 3 hours of posting these, the number 72 was found by a 3rd grader:  72=\frac{9!}{(5+1+1)!} .  Note the orange strip across the 72 above indicating that it had been found.  I’ve also had several families begin to play the game at home with their kids–exactly the kind of engagement I was hoping to stir up.

After being pressed by a few parents and students who didn’t believe all were possible, I spent a few hours exploring–I found all but one.  Here is my list of all of the missing numbers.  Obviously, this isn’t released to the students; it was for my own satisfaction.  But that leaves one big question:

Using only upper-elementary to middle-school comprehensible math functions, how do you make a 74 using only the digits 1, 9, 1, & 5.
(Pride bonus if you can use them in that order!)

Enhancing Creativity at Home

I had a great conversation last month with a Westminster colleague, Kay, as school was winding down.  While from completely different academic departments, our daughters are nearly the same age, and so we share lots of parenting ideas and stories.  Here’s some thoughts we developed to use with our kids this summer to enhance creative thinking and enhanced abilities in many areas.

Creativity in Literature:

Our daughters LOVE to read, so our first challenge to our girls would be for them to find a good stopping place 20-40 pages before the end of each book they read. Then we encourage them to describe how they think the story might end and why they think their hypothesis is possible.  If they see multiple ways it could end, they could describe alternates.  They should WRITE their predictions to get their minds to commit and more actively engage.  This need not be anything formal–but it can be if they like.  We just want the girls to think a bit more deeply about what they’re reading and engage.  Use what they know has already happened to draw conclusions about what might have happened.

After their informal (or formal) writing, read the remainder of the book and compare their prediction(s) to the author’s chosen end.  How close did they come to the author’s conclusion?  What information did the author use to end the book that they didn’t use?  Was the author’s conclusion reasonable? Was theirs?

There’s not any particular right or wrong here for us beyond getting our kids to think about what they’re reading and to engage the process.  We also hope they will become more attentive to details in their reading.


Allow the kids to help plan meals; help them understand the daily processes for planning nutritious meals.  Encourage them to participate in cooking, especially for anything you cook from scratch.  What choices do you make and why?

For cooking, talk to them about what the different ingredients do for the resulting dish.  Have them make predictions about what will happen when new ingredients are added or the collected ingredients are prepared.  To the best of your ability, explain what each step does and why it’s important to the final product.

Compare the predictions to the final results.  When some part inevitably turn out differently than predicted, learn why.  Take pictures along the way for you or your child to use when comparing hypotheses and outcomes.  When you come back to a recipe on another occasion, think about how it turned out last time and plan a change or an improvement.  Knowing what your child wants to happen, can she adapt the preparatory steps to accomplish that change?

In the end, this really isn’t all that different from the reading suggestions.  Engage, observe, explore, make predictions, and compare expected and actual results.  All along, use data to explain why you believe your claims are justified.


This is about where Kay & I ran out of time to chat that day, but we ended with the realization that what we were describing was exactly the scientific process, embedded in arguably “non-science” settings.  As I’ve mused over this for the past few weeks, I’ve realized that this process can be applied almost anywhere:

  • athletics (how do you get a better result? Why?),
  • mathematics (what kind of answer will I get? Can there be more than one? What will it look like?)
  • gardening (how do I get a particular plant to perform a specific way? Is that even possible with that plant?),
  • computer programming (getting a computer to do precisely what you ask of it), and so on.

Engage your children or your students. Get them to hypothesize and justify using data.  Teach them (and yourself) to be more alert to patterns and clues about past and future behavior.  Perhaps most importantly, determine if other outcomes are possible and what it would take to get there.  Envision something she or he hasn’t seen or done before and figure out what is needed to make it happen.  Then … make it happen.

Creativity explodes

Sometimes the lessons we teach keep on giving long after we think our classes are over. As you may know from my previous posts (here and here), I used an adaptation of the Four 4s activity to try to foster some creativity in my classes this Spring.

Basically, you are allowed four 4s, no more and no less, and no other digits, but any mathematical operations you want.  From this, create every integer from 1-100.  Of course, you can go far beyond that, but this seemed like a good place to start. Two of my students employed an unexpected approach, seeking ways to find all numbers 0-25 using a single four, thereby proving to themselves that the entire 1-100 table was solvable–often in multiple ways–without actually needing to write all the solutions.  It was a nice existence proof.

OK, most of their solutions were VERY complicated and the few they submitted ended up being sniped by others, but that wasn’t their goal. They wanted to know a solution was possible.  Two weeks ago, one of them, P, sent me an email describing how he leveraged his initial work to find solutions to the next 11 missing integers in Integermania’s Four 4s list.  Talking to P the next day, I found out that he accomplished all 11 in about half an hour.  To put this in perspective, I think Integermania has been running this list since early 2006.  Admittedly, most participants probably lost interest and stopped submitting entries long ago, but I still found P’s ability to find so many solutions so quickly to be pretty powerful.  The list has been designated “mostly inactive”; we’ll submit them anyway…

Here’s P’s unedited email (other than some LaTeX conversions by me).  If you need it, the functions he uses are detailed in the middle of my 2nd Four 4s post linked above.

Hi Dr. Harrow, I was in your room at about 4:00 and live about 30 minutes from school, got home, had a snack, opened integer mania. Once I scrolled through all the pages and found that the next missing number is 1138:

Realize 1140=3\cdot 4\cdot 5\cdot 19=4\cdot 15\cdot 19.

Using my list of 25, we can get 1140 using only three fours, then subtract 2 to get

1138=4\cdot \sigma(4!!)\cdot p_{4!!}-\sqrt{4}

1139=4\cdot \sigma(4!!)\cdot p_{4!!}-\Gamma(\sqrt{4})

1142=4\cdot \sigma(4!!)\cdot p_{4!!}+\sqrt{4}

1143=4\cdot \sigma(4!!)\cdot p_{4!!}+f_4

1159=4\cdot \sigma(4!!)\cdot p_{4!!}+p_{4!!}

Now, \displaystyle 1200=\frac{4!}{\left( \sqrt{4} \right) \%}, so

\displaystyle 1162= \frac{4!}{\left( \sqrt{4} \right) \%}-\sqrt{4}\cdot p_{4!!}

\displaystyle 1163= \frac{4!}{\left( \sqrt{4} \right) \%} - p_{\frac{4!}{\sqrt{4}}}

\displaystyle 1169= \frac{4!}{\left( \sqrt{4} \right) \%}-p_{T_4+\Gamma \left( \sqrt{4} \right) }

\displaystyle 1171=\frac{4!}{\left( \sqrt{4} \right) \%} -4! - p_{d(4)}

\displaystyle 1183=\frac{4!}{\left( \sqrt{4} \right) \%} - 4! + f_4

\displaystyle 1193=\frac{4!}{\left( \sqrt{4} \right) \%} - 4 + f_4

Which completes their list through 1200 (actually 1206). Just to emphasize how easy this is, find a way to get from 1200 to 1207 using only two fours. Oh man…

Thanks, P

PS: The highest exquisiteness level so far is the dude who made a googolplex (last page) with a 5.8. I might have gone a little past that…

Pretty impressive, I thought.  As for P’s exquisiteness, here’s what I computed

  • 1138 = Base 7 + 5 surcharges = Level 8.0
  • 1139 = Base 7 + 6 surcharges = Level 8.2
  • 1142 = Base 7 + 5 surcharges = Level 8.0
  • 1143 = Base 7 + 5 surcharges = Level 8.0
  • 1159 = Base 7 + 6 surcharges = Level 8.2
  • 1162 = Base 6 + 6 surcharges = Level 7.2
  • 1163 = Base 6 + 6 surcharges = Level 7.2
  • 1169 = Base 7 + 7 surcharges = Level 8.4
  • 1171 = Base 6 + 6 surcharges = Level 7.2
  • 1183 = Base 6 + 5 surcharges = Level 7.0
  • 1193 = Base 6 + 4 surcharges = Level 6.8

That these are pretty high levels relative to the rest of the list is totally irrelevant, in my opinion.  P has found a simple way to prove existence of a solution.  Often, a solution’s existence is enough to spur on investigation of more elegant answers.  P broke through.  Knowing that answers are possible, he challenges others to follow up with smoother results.

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_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.


\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.

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.




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.


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.