## Multiplication Sculptures and Sums of Cubes

A 15 by 15 multiplication sculpture built by Jane Kostick.

Take a multiplication table and build a column of cubes over each entry. For each column, use as many cubes as the product it sits upon. The result is a “multiplication sculpture” or “multiplication tower.” The picture shows a 15 by 15 multiplication sculpture built by Jane Kostick in 2008. For more examples, check out Maria Droujkova’s photo collection at moebius noodles.

Last spring, club members at Girls’ Angle built a 4 by 4 version out of cubes that were 3 inches on a side. They thought about its properties, such as how many cubes make up an N by N multiplication sculpture.

We’re going to address this last question and end up with a nice, concrete proof of a famous algebraic identity.

Over the xy entry, there are xy cubes, so we have to add up all products xy where x and y range over the values from 1 to N, and this totals

$(1+2+3+\dots+N)^2 = (\frac{N(N+1)}{2})^2$.

Wait a sec! Where else does that expression occur?

It is also the sum of the first N (positive) perfect cubes!

The implication is that the number of cubes sitting over the last row and column of the multiplication table must be $N^3$. If we can show this directly, we’d have a nice proof that the sum of the first N (positive) perfect cubes is equal to the square of the sum of the first N (positive) integers.

The columns of cubes over the last row of the multiplication table form a staircase with steps of height N. Saw these N columns off and get a flat, staircase-shaped plank, N cubes long and $N^2$ cubes high. Next, saw off the columns of cubes over the last column (of the multiplication table). You’ll get an almost identical staircase-shaped plank. The only difference is that the last step of height $N^2$ is gone because it was removed when the columns over the last row (of the multiplication table) were sawed off.

Turn one of these planks over, and the two planks will fit together perfectly to form an $N$ by $N^2$ rectangle, and an $N$ by $N^2$ rectangle has $N^3$ cubes in it!

So that gives a highly visual way to prove the identity

$1^3+2^3+3^3+\dots+N^3 = (\frac{N(N+1)}{2})^2$.

To read about how Jane made this 15 by 15 multiplication sculpture, and see hints about more of its properties, check out the December, 2008 issue of the Girls’ Angle Bulletin, pages 12-14 and 25-28.

## Girls’ Angle Bulletin, Volume 7, Number 6

The cover illustrates a neat result that Anna B. discovered and explains in this issue’s Anna’s Math Journal. She continued her investigation of paraboloids and discovered that orthogonal projection from a paraboloid coincides with the composition of stereographic projection and a special map M inspired by the optical properties of a paraboloid. For details, check out her column!

We also feature an interview with University of Minnesota assistant professor of mathematics Christine Berkesch Zamaere.

Next, Akamai Technologies computer scientist Kate Jenkins concludes her discussion of algorithms that find the “maximal subsequence” of a sequence.  Were you able to figure out an algorithm that determines the maximum subsequence of N numbers using O(N) computations? Kate’s article is just one example of how mathematics applies to problems in industry. In the past decades, so much information has been digitized, including books, pictures, video, weather, architectural plans, music, etc. Where there are numbers, there is the potential for mathematical analysis.

Emily and Jasmine return, this time designing star patterns for different numbers of states. We received positive feedback about their last project where they designed a stained glass window (see Volume 7, Number 4), so we plan to feature them more in the future. The two show how, with a bit of inquisitiveness, there’s mathematics.

We conclude with solutions to this summer’s batch of Summer Fun problem sets. Incidentally, if we had more room, we would have liked to include one more problem in the Summer Fun problem set on permutations. That problem set ended with a result of Zolotarev connecting the signs of certain permutations to the theory of squares modulo p, where p is a prime number. With more room, we’d have outlined Zolotarev’s proof of quadratic reciprocity using permutations. This proof is “just around the corner” from the material in the permutation problem set and Cailan’s Summer Fun problem set on quadratic reciprocity. As a challenge, you could try to reconstruct Zolotarev’s beautiful proof. Here’s a hint: The idea is to take a deck of pq playing cards, where p and q are distinct odd prime numbers. Consider the following 3 arrangements of the cards into a p by q rectangle:

Arrangement 1: Deal the cards out row by row, from left to right.

Arrangement 2: Deal the cards out column by column, from top to bottom.

Arrangement 3: Deal the cards out going along a NW-SE diagonal, with wraparound.

Consider the permutations defined in going from arrangement 1 to 2, from 2 to 3, and from 3 to 1.

We hope you enjoy our latest issue!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Marion Walter’s Theorem Via Mass Points

I recently had the good fortune of learning Marion Walter’s theorem from Marion Walter herself:

Marion Walter’s Theorem: In a triangle, draw line segments from each vertex to the trisection points on the opposite side. The six drawn line segments will form the edges of a central hexagon. The ratio of the area of the hexagon to that of the whole triangle is 1/10.

An efficient way to prove Marion Walter’s theorem is to use mass points.

In this post, I’ll give details because the proof is a model example of the mass points technique. If you’re having difficulty learning the technique, I hope this post will help it all come together for you. As always, try to use mass points to prove the theorem yourself, and, only after you have tried, read on. If you’ve never heard of mass points before, google it or check out Volume 7, Number 3 of the Girls’ Angle Bulletin.

## Girls’ Angle Bulletin, Volume 7, Number 5

A paraboloid of revolution adorns the cover. Anna investigates cross sections of paraboloids in this issue’s Anna’s Math Journal. It sure feels like Anna is embarking on an interesting mathematical journey with this new topic. We hope you’ll be inspired to follow-up on her work.

However, first up is the concluding half of our interview with University of Oregon Professor Emerita Marie Vitulli. Read some of the ways she thinks gender bias in mathematics can be countered.

Next, Akamai Technologies computer scientist Kate Jenkins discusses algorithms that find the “maximal subsequence” of a sequence.  Her first part closes with an interesting challenge. Can you find a solution to her challenge before she gives it in the next issue?

Brit Valeria Golosov presents a fictional account of how she imagines that Brahmagupta derived his famous formula for the area of a cyclic quadrilateral. Valeria is entering her final year of secondary school in London.

This issue’s Math In Your World was specifically requested by Vida John. We love receiving content requests from members and subscribers. This Bulletin is written for members and subscribers and members and subscribers are welcome to control Bulletin content by emailing us comments and suggestions.  Please don’t be shy about emailing us about anything to do with math! We also welcome and encourage all members and subscribers to send in solutions to the Summer Fun problem sets. We might even publish your solution in the Bulletin.

This summer’s batch of Summer Fun problem sets address magic squares, mass points, quadratic reciprocity, and permutations. Contributors include Johnny Tang and Cailan Li, both recent high school graduates who will be heading to college this coming fall. The central theme of Volume 7, Number 3 of the Bulletin was the concept of center of mass which underlies the technique of mass points. In that issue, we didn’t have enough room to include many problems to practice the technique. So that’s one reason why we included a problem set on mass points. The problems range from introductory level to some that will hopefully entertain those experienced in the technique. Cailan’s problem set takes readers from the rudiments of modular arithmetic all the way through a proof of Gauss’s Law of Quadratic Reciprocity, following a proof by D. H. Lehmer. The set on permutations culminates with a result of Zolotarev that links signs of certain permutations to the Legendre symbol introduced in Cailan’s problem set.

To whet your appetite, suppose AB, and C are the angle of a triangle. Can you prove that

$9 > 3 + 2(\cos 2A + \cos 2B + \cos 2C) \ge 0$

with equality if and only if the triangle is equilateral? For a spiffy way to prove this, check out the Summer Fun problem sets!

We conclude with a brief account of a wonderful field trip we took to MIT’s Department of Aeronautics and Astronautics, which was generously organized by Professor Karen Willcox,

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Cubes in one-point perspective

This post is a reply to Jamie’s comment on Drawn to Math:

Would anyone know how to construct a cube in one point perspective?

This is an excellent question because it isolates an important, simplified situation that enables one to study key aspects of perspective drawing.

(If you are completely new to perspective drawing, I’d suggest working through the Summer Fun problem set on pages 21-22 of the June, 2013 issue of the Girls’ Angle Bulletin.)

## Girls’ Angle Bulletin, Volume 7, Number 4

The cover features a planar configuration by Leah Berman, associate professor of mathematics at the University of Alaska Fairbanks.  There are 240 lines and 240 points arranged so that each line contains 6 of the 240 points and each point sits on 6 of the 240 lines. More images of planar configurations by Leah and Nadine Alise can be found in Mathematical Buffet.

This issue’s interview is with University of Oregon Professor Emerita Marie Vitulli.  In this first part of a 2 part interview, we learn about Prof. Vitulli’s field and how she got into mathematics.

Special thanks to Professor Emeritus Thomas Moore for contributing a problem about Pythagorean triples.  If you haven’t heard of Pythagorean triples, Prof. Moore gives a brief introduction and more challenges in Pythagorean Triples Challenge.

Angles pervade much of this issue.  In Learn By Doing, Addie Summer covers the basic of angle measure.  Then, Lightning Factorial follows Emily and Jasmine as they use angles to design a stained glass window.  Finally, in this issue’s Math In Your World, I write about one of John and Jane Kostick’s latest creations, which they dub the Quintetra Assembly.  To compute the necessary angles needed to create a wood sculpture for this amazing polyhedron, several angles must be computed.  In this article, I sketch how to determine these angles and include a net, courtesy of the Kosticks, for Jane’s Quintetra block, 30 of which can be used to build a model of the Quintetra Assembly.

Finally, Anna continues her investigation of x to the x.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Girls’ Angle Bulletin, Volume 7, Number 3

Much of this issue is about the center of mass, including Julia Zimmerman’s cover drawing which features a mobile fantasy. In Math In Your World, we cover the basics involved in ensuring that a mobile will balance properly. You could use this knowledge to get your pet hamster to lift your car. In Center of Mass, we explain the “piecemeal” property of the center of mass: that you can compute the center of mass part by part.  This property forms the basis of the problem solving technique known as “mass points.”

To understand the center of mass and torque, it helps to learn about vectors.  So Robert Donley, a.k.a. Math Doctor Bob, introduces vectors in his second installment of Learn by Doing.  If you work through the problems in his column, you’ll learn about vector addition, scalar multiplication, and the dot and cross products.  Because Bob wanted to lay the groundwork for understanding physical vectors such as force and torque, he confined himself to discussing vectors over the real numbers.

In our interview, meet University of Michigan Professor of Mathematics Karen Smith.   Prof. Smith discusses a wide range of topics, including math that interests her, how she goes about solving math problems, and how she got into math.  Her route to math is rather exceptional in that after graduating from Princeton with a Bachelor’s degree, she taught in the school system before going to math graduate school.  She is a recent Clay Scholar and she contributed an autobiographical essay to the book Complexities: Women In Mathematics, edited by Bettye Anne Case & Anne M. Leggett.  In this interview, she even describes an unsolved problem which middle school students could begin to explore.

In Anna’s Math Journal, Anna launches into an exploration of the function $f(x) = x^x$ after being asked the following question: For what number x other than 1/2 is $x^x = (\frac{1}{2})^{1/2}$?

“You” helps 3/7 find a nifty solution to a curious dilemma in Coach Barb’s Corner…involving fractions, of course!

Finally, if you like dissection problems, see if you can find a nice way to tile a regular hexagon into pieces that can be rearranged to form an equilateral triangle. Henri, a student at the Buckingham, Browne, and Nichols middle school in Cambridge, was presented with just this challenge, and we exhibit his beautiful solution inside.  This problem is an instance of the Wallace-Bolyai-Gerwien theorem which says that any polygon can be dissected into parts that can be rearranged to form any other polygon of the same area. Even though there is a general construction to create such tilings, there’s still the challenge of finding nice ones, and Henri’s is quite elegant.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!