Monthly Archives: May 2013

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.

Multiplication Practice Plus Creativity

I hope this post is particularly helpful for parents and teachers of elementary school children.  Through my Twitter network last week I found via @Maths_Master‘s Great Maths Teaching Ideas ‘blog a 2010 post summarizing Dan Finkel’s Damult dice game. Recognizing that “practicing times tables can be unmotivated and boring for kids,” Damult is an attempt to make learning elementary multiplication facts more entertaining. I offer some game variations and strategies following a description of the game.

Here’s Dan’s game:

image via Wikipedia

Each player takes turns rolling 3 dice. First to break 200 (or 500, etc.) wins. On your turn, you get to choose two dice to add together, then you multiply the sum by the final die. That’s your score for that turn.

Simple; no bells, no whistles. For example, I roll a 3, a 4, and a 6 on my turn. I could either do (3+4) times 6 for 42 points, OR (3+6) times 4 for 36 points, OR (4+6) times 3 for 30 points. I’ll take the 42 points.

I spent some time playing this with kids the other day and I saw that (1) it was genuinely fun, and (2) it gives you almost all the multiplication practice you could ask for. In fact, it gives even more, because the choice of which dice to add and which to multiply reveals some interesting structure of numbers. Seriously, get a kid hooked on this game, and it’s the equivalent of dozens or hundreds of times table practice sheets.

It’s a fun activity idea by itself.  Damult combines a bit of luck and memory, and rewards the ability to recall multiplication facts.  As an added bonus, it requires players to be able to manipulate objects in their heads–how many different ways can the three given dice be manipulated in summation stage to create unique products? How can a player ensure that she has found the biggest product for her score?  Try the game!


This is a great opportunity for parents to engage with their  children as they learn multiplication facts.  Parents and teachers could play along, or the learner might be the only player, talking out loud so that the teacher or parent can “hear the thinking.”

I love that the game completely randomizes the multiplication tables.  This significantly enhances recall as memory is not tied to particular patterns or positions on fact pages. Players must adapt to each random roll.

In any variation, there obviously should be a discussion among all players about what products were found to confirm the results. Make the game more formative or more competitive, depending on the experience level of the players.  In more competitive variations with experienced learners, if a product was miscalculated and claimed, you might decide that no score should be recorded for that round.

If you’re guiding someone on this it is critical that you DO NOT give answers.  Students need to explore, hypothesize, discover errors, learn how to communicate their conclusions in clear and concise language, and to learn how to defend their findings while also learning how to admit flaws in their reasoning when faced with contradicting data.  Experimenting and discovery is always deeper, richer,and more long-lasting than just being told.  Remember the Chinese Proverb: “I hear and I forget. I see and I remember. I do and I understand.” Always seek understanding.

The first comments on Dan’s post noted that while one player was summing and multiplying, the other player(s) were largely disengaged. Also, the game could drag on as unconfident players tried to make sure  they had explored every possibility.  To address that and several other possibilities, I offer the following Damult variations.  Some more complex variations are toward the end. Read on!

Finally, if you’ve read my ‘blog much, you know that I’m a huge fan of leveraging technology for math learning, but this is one of those situations where I think you should 100% unplug. To learn multiplication facts is to learn some of the basic grammar and vocabulary that makes the language of mathematics work.  You simply can’t communicate mathematically with an underlying awareness of how the structure of the language works.


Variation 1: Adding a timer to the game could cure the slow-down issue. Depending on the age of the child and his/her familiarity with multiplication, the timer can be longer or shorter.  If the skill levels of the players are unequal, make the timer unequal.  (I love the adage, “Fair is seldom equal, and equal is seldom fair.”)

Variation 2: Why must only one player be active? The players could take turns rolling the dice while both record scores based on what they find.  If a particular combination was not noticed by one player, that player doesn’t get to consider it for his/her score.

Variation 3 – As an aside, notice that Dan implicitly claims there are only 3 possible sums from a 3-dice roll. Will that always be the case?  Can you convince someone why your solution is correct?  

(For 3 dice the maximum number of possible sums is 3. When and why would there be fewer products?)

Variation 4 – How many multiplication facts are possible using only 3 dice?

This would be a great number sense exploration.  Some may try it by gathering lots of data, others may have more sophisticated reasoning.  I suggest that you or your students hypothesize an answer first along with some reason why you think your hypothesis is correct.  Different answers are OK, and you can always revise your hypotheses if you get evidence leaning in another direction. No matter what, have fun exploring and learning. 

(Middle School extension: Damult creates products of axb where a can be any integer 1 – 6 and b can be any integer 2 – 12.  That gives 66 different products if you count different arrangements (3×4 and 4×3) as different products. Can you or your student see why? How many outcomes are possible if you look only at the product result and not at the factors which created it?)

Variation 5 – After discovering or just using the answer to the last variation, you could use a table of multiplication facts and see how quickly different facts and be “discovered” from rolls of the dice.  After rolling 3 dice, mark off all multiplication facts you can using the sum-then-multiply combination rules posed at the beginning.  This might be a fun way for early learners to familiarize themselves with multiplication patterns.

NOTE: If you play variations 4 or 5 as a game, you’ll likely want (or need) to stop before all possibilities are found.  Some (eg, 6×12 and 1×2) will be pretty uncommon from dice rolls.

Variation 6 – You could make a Bingo-like or a 4 or 5-in-a-row game.  The first person to mark off a certain number of facts or the first to get a certain number in a row would be a winner.

Variation 6 – If you try the last few variations, you’ll see that some products occur much more frequently from the dice rolls than others.  This could be used to introduce probability. Which products are more likely and why?

As an example, I suspect 3×7 could happen six times more often 1×2.  Can you convince yourself why 3×7 is so much more likely?  Can you see why 3×7 is exactly six times more likely than 1×2?

Variation 7 – Why restrict yourself to 3 dice? When just starting out, using more than 3 dice would definitely be a frustration factor, but once you’ve got a good grip on the game, consider rolling 4 dice and allow players to multiply the sum of any 2 or 3 of the dice by the sum of the remaining dice.

By my computation, using 4 dice means there are up to 7 possible combinations in a given roll.  Can you prove that? Being able to consistently find them all is likely to be a very difficult challenge, but it is a phenomenal and early opportunity to stretch a young person’s mind into considering multiple outcomes and reliable ways to guarantee that you’ve considered all possibilities.

Variation 8 – Why go for maximum products and being the first to get to 200 or 500 points?  Why not try for a low score (like golf), seeking minimum products  and being the last to exceed 100 or 200?

Variation 9 – Stealthy Calculus:  OK, my analysis on this one goes way deeper than is necessary to play the game, but sometimes knowing more than is necessary can give insights and can help you lead others toward developing “math sense”–a truly invaluable skill.

LOW LEVEL – After you’ve played this a few times, ask the player(s) if there is some strategy that could be used to guarantee the biggest (or smallest) possible product for any roll.  This could be a great mathematical experiment for which the solutions are not at all intuitive, I think.  Some might figure it out quickly and others might need to gather lots of data, comparing products from lots of rolls before distilling the relationship.

If you’re guiding someone on this it is critical that you DO NOT give the answer.  Students need to explore, hypothesize, learn how to communicate their conclusions in clear and concise language, and to learn how to defend their findings while also learning how to admit flaws in their reasoning when faced with contradicting data.  If you don’t know the answer, stop reading now and figure it out for yourself. I provide an answer in the next paragraphs, but experimenting and discovery is always deeper, richer,and more long-lasting than just being told.  Remember the Chinese Proverb: “I hear and I forget. I see and I remember. I do and I understand.” Always seek understanding.

MUCH HIGHER LEVEL – As a calculus teacher, the very first fact that struck me was Damult’s implied goal: Getting the largest possible product from any roll of three dice.  That’s an optimization problem, and I knew from calculus that the greatest possible product of two numbers whose sum was constant happens when the two numbers are as close as possible to being equal.  Likewise, the smallest possible product happens when the two factors are as far apart as possible.  (If you recall some calculus of derivatives, I encourage you to prove these for yourself.  If anyone asks, I could write a future post with the proof.)

In Dan’s initial example above in which 3, 4, and 6 were rolled, I stopped reading after the first sentence of paragraph 2 (pausing to think and draw your own conclusions is a great habit of the mind) for a few moments as I thought, “I know 3+4 and 6 are as close to equivalent as I can get, so 7*6=42 is the greatest possible product.”  I didn’t even look at the other possibilities, I knew they were less. This fact was established (unnecessarily for me) in the end of the paragraph.

Without calculus, I propose students try making tables of their data.  They’ll have up to three unique products (Variation 3) and will need to explore the data before hopefully discovering the relationship. If a young person doesn’t discover the relationship, Don’t tell him/her! it is far better to leave a question as unanswered to think on and answer another day than to have a relationship given unearned.  Value comes from effort and discovery. Don’t cheat young learners out of that experience or lesson.

Conclusion: Don’t just play a game. Be creative! Strategize! Encourage young ones not just to play, but to play well. Children are quite creative in free play as they continually make new and adapt old “rules”.  Why should intellectual play be any different?  I’d love to see what variations others discover or have to offer.

A Teacher’s Prayer

At the end of this week, I’m leaving Westminster, my school of 23 years to pursue a phenomenal opportunity for my family and me at the Hawken School in Cleveland, OH.  As you can imagine, my farewells this past week and now into my last have been both bitter and sweet.  In an attempt to craft a fitting acknowledgement for the gratitude and respect for all I have learned here, I penned a Teacher’s Prayer inspired by a prayer accredited to and adapted by Mother Teresa.  I hope you find something of value in it.


A Teacher’s Prayer

My students are often scared they’re not going to understand.  They fear failure and disappointment.  Even when they hold back, help me continue to encourage and challenge them anyway.

Sometimes my students don’t do all they can to maximize their learning.  Help me continue to meet them where they are anyway.

Some of my students don’t care for my classes’ content and my lessons might not be well received. Help me continue to be passionate about every moment anyway.

Students sometimes ask questions I believe have already been answered. Help me continue to always address them with deep respect anyway.

Sometimes none of the explanations I know make an idea stick.  Even when I already may have offered enough, help me continue to graciously find another approach anyway.

If my lessons break free from where I’ve planned, some may not be able to keep up, and others may outrun me or even catch me in a mistake.  Help me continue to always take chances and run with them anyway.

Even when I think I know every connection and approach a student might offer, help me continue to listen to every “new” idea as if it were the best insight any person has ever made, because to the one brave enough to share, it was.

New technologies and teaching ideas threaten how and what I teach; new tools may force me to redevelop or abandon favorite lessons I’ve spent years perfecting.  But a new idea also may be just what I need to reach someone who’s been beyond my grasp, so help me continue to embrace innovation anyway.

I sometimes despair I won’t be able to reach the ambitious goals I set for my students or me.  Even though I will sometimes fall short, help me continue to be ambitious anyway.

Grant me strength to give more than I think I have, determination to overcome what I didn’t think I could achieve, a love deeper than I believe exists, a hunger to make the world a better place even if only in a small way and against all odds, grace to learn from my inevitable mistakes, and courage to never fear growth.  After all, that’s what we’ve always asked of our students anyway.  Amen.

Wolakota, Chris


I leave you with a favorite song:  Lee Ann Womack’s I Hope You Dance.

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.

Gender and Expectations Lessons from Research

A couple reports from NPR yesterday have me thinking about some of the articles I’ve compiling in my Diigo library about what I’ve learned about gender and expectations differences in parenting and teaching.  I don’t have anything particular to tie together here, but I thought these four resources were more than I could comfortably tie together in a coherent Tweet thread, so I thought I’d gather them into an impromptu ‘blog post.

Girls May Get More ‘Teaching Time’ From Parents Than Boys Do via @NPR.  Excerpts:
… ” ‘How often do you read with your child?’ or ‘Do you teach them the alphabet or numbers?’ … Systematically parents spent more time doing these activities with girls.”
… “Since parents say they spend the same amount of time overall with boys and girls, Baker’s analysis suggests that if parents are spending more time with girls on cognitive activities, they must be spending more time with boys on other kinds of activities.”
… “The costs of investing in cognitive activities is different when it comes to boys and girls. As an economist, he isn’t referring to cost in the sense of cash; he means cost in the sense of effort.”

Gender Gap Disappears in School Math Competitions via .  Excerpt:
… “Most school math contests are one-shot events where girls underperform relative to their male classmates. But a new study by a Brigham Young University economist presents a different picture.  Twenty-four local elementary schools changed the format to go across five different rounds. Once the first round was over, girls performed as well or better than boys for the rest of the contest.”
… “It’s really encouraging that seemingly large gaps disappear just by keeping [girls] in the game longer.”

A broader look at school expectations leading to enhanced math performance:  What Distinguishes a Superschool From the Rest via .
… “The difference seems to lie in whether a school focuses on basic competence or encourages exceptional achievement. While almost all the schools saw it as their responsibility to cover the math knowledge necessary to do well on the SATs, the authors noted that “there is much less uniformity in whether schools encourage gifted students to develop more advanced problem solving skills and reach the higher level of mastery of high school mathematics.”
… “The fact that the highest achieving girls in the U.S. are concentrated in a very small set of schools, the authors write, indicates ‘that almost all girls with the ability to reach high math achievement levels are not doing so.’ ”

Girls, Boys And Toys: Rethinking Stereotypes In What Kids Play With via @NPR.  Excerpts:
… Some toy companies are re-thinking gender-specific marketing and branding.
… “I think what they were worried about was causing gender identification needlessly — to turn off passive learning, passive expression down the road, even passive economic opportunity for girls or boys if they felt they couldn’t do something because of societal norms,”
… “It’ll be interesting to see how this changes the attitudes of parents and of kids over time or whether or not it does. There may be some hard-wired differences,”


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.