Tag Archives: “four 4s”

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.

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.

In 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.

The 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!)

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_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.

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.