 # Mathematics Galore!: The First Five Years of the St. Mark’s Institute of Mathematics

James Tanton
Edition: 1
Pages: 289
https://www.jstor.org/stable/10.4169/j.ctt5hhb20

1. Front Matter
(pp. i-viii)
(pp. ix-x)
3. Introduction
(pp. xi-xvi)

Eight years ago I took the plunge: I left the college and university world and became a high school mathematics teacher. It was time for me to do what I preached.

For many years I conducted workshops for educators that I thought were innovative, exciting and spoke directly to the mathematical experience. My goal was to enhance joyful learning for students through joyful work with their teachers and I designed activities that illustrated, I hoped, the value of intellectual play, of discovering and owning ideas, of questioning assumptions, of flailing, and of experiencing success. Although deemed interesting and fun my...

• 1 Arctangents
(pp. 1-10)

The following diagram starts with a right isosceles triangle with legs 1 and stacks an additional right triangle with a leg of 1 onto the hypotenuse of a previously constructed right triangle.

a) Ifnright triangles are stacked in this way, what is the length of the longest line segment in the diagram?

b) If we keep stacking right triangles, will the diagram ever make a full turn of rotation? Two full turns of rotation?

The angle a line of slopemmakes with the horizontal is called thearctangentof the slope, denoted arctan (m). For example, a...

• 2 Benfordʹs Law
(pp. 11-18)

Here is one of my favorite mathematical mysteries.

Consider the powers of two:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,

Do any of them begin with a 7?

If so, which is the first power of two that does? What is the second, and the third? Are there ten powers of two that begin with a seven? Are there infinitely many?

If, on the other hand, no power of two begins with a seven, why not?

The powers of three begin

1, 3, 9, 27, 81, 243, 729, 2187, 6561, …...

• 3 Braids
(pp. 19-24)

The language of ABABA uses only two letters, A and B, and any combination of them is a word. (Thus, for instance, ABBBBABAA and BBB are both words.) Also, strangely, a blank space, , is considered a word.

The language has the property that any word that ends in ABA or in BAB has the same meaning as the word with them deleted. (Thus, for example, BBABA and BB are synonyms.) Also, any two consecutive As or Bs can be deleted from a word without changing its meaning. (Consequently, BAABBBA, BAABA, BBA, and A are equivalent words.)

Warm-up. Show that...

• 4 Clip Theory
(pp. 25-30)

This is one of my favorite teasers. I use it in all of my extracurricular courses for teachers and students. Certainly patterns must be true!

Here we are assuming that the dots are spaced so that the maximal number of pieces appears when each pair of dots is connected by a straight line.

Going Further. Seven dots? Eight dots? Nine dots?

Bending the Rules. This month’s puzzler is a surprise. (Really do try it!) A greater surprise still lies hidden in the mathematics if one is willing to be flexible in one’s play, so flexible as to flex the lines!...

• 5 Dots and Dashes
(pp. 31-36)

1. The sequence of square numbers begins 1, 4, 9, 16, 25, … and thenth square number isn².

The sequence of non-square numbers begins 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, ….

a) What’s the 100th non-square number?

b) Find a formula for thenth non-square number

2. The sequence of triangular numbers begins 1, 3, 6, 10, 15, … and thenth triangular number is$\frac{1}{2}n(n+1)$.

The sequence of non-triangular numbers begins 2, 4, 5, 7, 8, 9, 11, …

a) What’s the 100th non-triangular number?

b) Find a...

• 6 Factor Trees
(pp. 37-42)

In grade school students draw factor trees. Here is a tree for 36,000:

At each stage we split the number at hand into a pair of factors, halting at the primes. (This forces the tree to stop. Good thing that 1 is not considered prime!) The tree allows us to write the starting number as a product of primes. Here we see 36000 = 2 · 2 · 2 · 2 · 2 · 3 · 3 · 5 · 5 · 5. What is astounding (though most people don’t seem to think so) is that despite possible different choices...

• 7 Folding Fractions and Conics
(pp. 43-52)

Sally draws a straight line from the bottom left corner of a blank piece of paper. She challenges Terrell to make a crease in the paper that bisects the angle formed (that is, cuts that angle exactly in half). “Easy” says Terrell as he lifts up the bottom edge, aligns it with the straight mark, and folds.

“Now,” chortles Terrell in response, “I challenge you to maketwocrease marks that divide your angle exactly into thirds.”

It can be done using nothing more than creases in the paper. How?

Without the aid of any tools it is very difficult...

• 8 Folding Patterns and Dragons
(pp. 53-62)

Take a strip of paper and imagine its left end is taped to the ground. If we pick up the right end, fold the strip in half, and unfold it, the paper has a valley crease in its center. Label it 1.

Suppose we perform two folds, lifting the right end up over to the left end. When we unfold the paper, we find three creases: two valley creases and one mountain crease. Label the mountain crease as 0.

For three folds we obtain

The sequence for four folds turns out to be 110110011100100.

a) What is the sequence of...

• 9 Folding and Pouring
(pp. 63-68)

One gallon of water is distributed between two containers labeled A and B. Three-quarters of the contents of A are poured into B, and then half the contents of B are poured back into A.

This process of alternately pouring from A to B (three-quarters of the content) and then from B to A (half the content) is repeated.

What happens in the long run?

Here’s something to try:

Take a strip of paper and make a crease mark at an arbitrary position.

Make a new crease halfway between the position and the left end of the strip by folding...

• 10 Fractions
(pp. 69-76)

a) The following are true:$\frac{26}{65}=\frac{2}{5}\quad\quad \frac{266}{665}=\frac{2}{5} \quad\quad \frac{2666}{6665}=\frac{2}{5}.$

Does 2 followed bynsixes divided bynsixes followed by 5 always equals 2/5?

b) We also have$\frac{49}{98}=\frac{4}{8}\quad\quad \frac{499}{998}=\frac{4}{8}\quad\quad \frac{4999}{9998}=\frac{4}{8}.$

Does 4 followed bynnines divided bynnines followed by 8 always equals 4/8?

c) Find another example.

(This material is adapted from Chapter 11 ofTHINKING MATHEMATICS! Volume 1: Arithmetic=Gateway to All.)

Suppose we wish to share 7 pies among 12 boys.

We could divide each pie into 12 parts and give each boy 7 pieces. But let’s be practical as were the ancient Egyptians (ca. 2000 b.c.e.)....

• 11 Integer Triangles
(pp. 77-86)

What property do each of the following figures share?

Find another right triangle (with integer sides) with the property. Is there another integer rectangle with the property?

Heron’s Formula. In 100 c.e. Heron of Alexandria (also known as Hero of Alexandria) published a remarkable formula for the area of a triangle in terms of its three side-lengths$\text{area=}\frac{1}{4}\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}$

Thus the areaAof the 15-20-7 triangle above is$A\text{=}\frac{1}{4}\sqrt{(42)(2)(28)(12)}=\frac{\sqrt{28224}}{4}=42.$

Proving Heron’s formula is not difficult conceptually. (The algebra required, on the other hand, is a different matter!) Here are two possible approaches:

Proof 1. Draw an altitude and label the...

• 12 Lattice Polygons
(pp. 87-94)

This chapter might drive you dotty! As you will see, we’re asking a number of tricky questions about polygons on a square lattice of dots. There are many interesting observations to be made about them. Nothing requires high-powered mathematics, just, perhaps, high powered ingenuity! Have a piece of graph paper at your side as you ponder these tricky proposals.

Is it possible to draw an equilateral triangle on a square array of dots so that each corner of the triangle lies on a dot?

We can draw squares of areas 1, 2, 4, 5, and 8 on a square lattice:...

• 13 Layered Tilings
(pp. 95-102)

Here are two tilings of a 6 × 6 grid of squares using 1 × 2 tiles (“dominos”).

Each tiling is structurally flawed in the sense that each possesses a line along which it could slide.

Devise a tiling of a 6 × 6 grid that is structurally sound.

Is there a structurally sound tiling of an 8 × 8 board?

The following 2 × 3 board is double tiled with six dominos, meaning that each cell of the grid is covered by exactly two dominos.

We’ll say that a 2 × 3 grid is 2-tilable with dominos. It is...

• 14 The Middle of a Triangle
(pp. 103-110)

Most everyone would agree that the point of intersection of the two diagonals of a rectangle deserves to be called the “middle” of the rectangle. After all, it would the balance point of the shape if the rectangle were cut from uniform material and placed horizontally on the tip of a pencil. So maybe “middle” means “balance point.”

Challenge. Three squares are adjoined to make an L-shape piece. Find the location of the balance point of this figure. Does it seem reasonable to call this the “middle” of the L-shape?

Design an L-shaped piece whose balance point lies outside the...

• 15 Partitions
(pp. 111-122)

Apartitionof a counting numberNis an expression that representsNas a sum of counting numbers. For example, there are eight partitions of 4 if order is considered important:$\begin{matrix} 4\quad 3+1 \\ 2+1+1 \\ \end{matrix}\quad \begin{matrix} 1+3 \\ 1+2+1 \\ \end{matrix}\quad \begin{matrix} 2+2 \\ 1+1+2 \\ \end{matrix}\quad \begin{matrix} {} \\ 1+1+1+1 \\ \end{matrix}$

There are five unordered partitions of 4:$4\quad 3+1\quad 2+2\quad 2+1+1\quad 1+1+1+1+1$

a) How many ordered partitions are there of the numbers 1 through 6. Any patterns?

b) How many unordered partitions are there of the numbers 1 through 6. Any patterns? Will these partition numbers continue to be prime?

One can place restrictions on the types of partitions one wishes to count. For example, there are eight partitions of 10...

• 16 Personalized Polynomials
(pp. 123-128)

The product of any two consecutive integers is divisible by 2. This is obvious since one of them must be even.

Prove that the product of any three consecutive integers must be divisible by 6. (For example, 7 × 8 × 9 = 504 is a multiple of 6.)

Prove that the product of any four consecutive integers is divisible by 24, the product of five consecutive integers by 120, and the product of six consecutive integers by 720. Does this remain true even if some of the consecutive integers are negative?

[By the way … What are the numbers...

• 17 Playing with Pi
(pp. 129-136)

This puzzler is a classic:

A rope fits snugly around the equator of the Earth. Ten feet is added to its length. When the extended rope is wrapped about the equator, it magically hovers at uniform height above the ground. How high off the ground?

A second rope fits snugly about the equator of Mars and ten feet is added to its length. How high off the ground does this extended rope hover when wrapped about the planet’s equator?

A third rope fits snugly about the (tiny) equator of a planet the size of a pea. When ten feet is...

• 18 Pythagorasʹs Theorem
(pp. 137-144)

A triple of positive integers (a,b,c) is called aPythagorean tripleif${{a}^{2}}+{{b}^{2}}={{c}^{2}}$.

Did you know that an ordinary multiplication table contains such triples?

Choose two numbers on the diagonal (these are square numbers) and two numbers to make a square. Sum the two square numbers, take their difference and sum the other two numbers. You now have a Pythagorean triple!$\begin{array}{lll} a=25-4=21 \\ b=10+10=20 \\ c=25+4=29 \\ \end{array}\quad \Rightarrow \quad {{20}^{2}}+{{21}^{2}}={{29}^{2}}$

Choosing 36 and 1 gives$\begin{array}{lll} a=36-1=35 \\ b=6+6=12 \\ c=36+1=37 \\ \end{array}\quad \Rightarrow \quad {{12}^{2}}+{{35}^{2}}={{37}^{2}}$

Question 1. Which two square numbers give the triple (3, 4, 5)? Which give (5, 12, 13) and (7, 24, 25)?

Question 2. Why does this work?

Tough Challenge....

• 19 On Reflection
(pp. 145-156)

A popular discovery activity in the middle school curriculum is

A ball is shot from the bottom left corner of a3 × 5billiard table at a 45 degree angle. The ball traverses the diagonals of individual squares drawn on the table, bouncing off the sides of the table at equal angles. Into which pocket, A, B, or C, will it eventually fall?

Experiment with the tables pictured below. What do you notice about those tables that have the ball fall into the top-left pocket A? Into the top-right pocket B? Into the bottom-right pocket C? Test your theories...

• 20 Repunits and Primes
(pp. 157-162)

Arepunitis a number all of whose digits are one:

1, 11, 111, 1111, 11111, ….

Some repunits are prime (such as 11 and 1111111111111111111) and others are composite. (What are the factors of 111, of 1111 and of 11111?)

It is generally believed, but not proven, that infinitely many repunits are prime. (Only five prime repunits are known: those with 2, 19, 23, 317 and 1031 digits.) Your challenge is to establish that

If a repunit is prime, then the number of its digits is prime.

A number isprimeif it has exactly two distinct factors (whole...

• 21 The Stern-Brocot Tree
(pp. 163-170)

Here is something fun to think about. Consider the fraction tree:

It is constructed by the rule

Each fraction has two children: a left child, a fraction smaller than 1, and a right child larger than 1.

a) Continue drawing the fraction tree for another two rows.

b) Explain why the fraction$\frac{13}{20}$will appear in the tree. (It might be easier to first figure what should be the parent of$\frac{13}{20}$, its grandparent, and so on.)

c) Might the fraction$\frac{13}{20}$appear twice in the tree?

d) Will the fraction$\frac{457}{777}$appear in the tree? Might it appear...

• 22 Tessellations
(pp. 171-182)

A polygon is said totessellatethe plane if it is possible to cover the entire plane with congruent copies of it without overlap (except along the edges of the figures). The tessellation is called atilingif each edge of one polygon matches an entire edge of an adjacent polygon.

Parallelograms tile the plane. As two copies of the same triangle can be placed side-by-side to form a parallelogram we see thatevery triangle tiles the plane.

1. The tiling with triangles shown isperiodic, meaning that it possesses translational symmetries in (at least) two non-parallel directions. That is, it...

• 23 Theonʹs Ladder and Squangular Numbers
(pp. 183-192)

Everyone knows that$\sqrt{2}$is irrational.

a) Prove it! That is, prove there is something mathematically wrong in writing$\sqrt{2}=a/b$for some integersaandb.

b) Prove that$\sqrt{3}$is irrational.

c) Prove that$\sqrt{6}$is irrational.

d) Prove that$\sqrt{2}+\sqrt{3}$is irrational.

e) That$\sqrt{2}$can be written as${2^{1/2}}$shows that it is possible to raise a rational number to a rational power to obtain an irrational result. Is it possible to raise an irrational number to an irrational power to obtain a rational result?

The square numbers begin 1, 4, 9, 16, 25, … We can...

• 24 Tilings and Theorems
(pp. 193-204)

1. Here’s a picture of a portion of bathroom floor tiled with three differently shaped tiles: a big square, a small square, and a parallelogram.

Place a dot at the center of each white square. (Really. Please do it!) Do you see from this we get an easy proof of the following remarkable result from geometry?

If squares are drawn on each edge of a parallelogram, their centers form a perfect square.

2. Pythagoras’s theorem states that if squares are drawn on the sides of a right triangle, then the sum of areas of the small squares equals the area of the...

• 25 The Tower of Hanoi
(pp. 205-210)

The Tower of Hanoi puzzle consists of three poles on which a collection of differently sized discs sit on one pole in order of size, largest on the bottom. The challenge is to transfer all the discs to a different pole in such a way that only one disc is moved at a time and no disc throughout the process ever sits on top of a disc of smaller size.

The two-disc version of the puzzle is easy to solve. It takes three moves.

a) Solve the three-disc version of the puzzle. How many moves does it take?

An eight-disc...

• 26 Weird Multiplication
(pp. 211-220)

Here’s an unusual means for performing long multiplication. To compute 22 × 13, for example, draw two sets of vertical lines, the left set containing two lines and the right set two lines (for the digits in 22) and two sets of horizontal lines, the upper set containing one line and the lower set three (for the digits in 13).

There are four sets of intersection points. Count the number of intersections in each and add the results diagonally as shown:

There is one caveat as illustrated by the computation 246 × 32:

5. ### Appendices

• Appendix I Numbers that are the Sum of Two Squares
(pp. 221-230)
• Appendix II Pickʹs Theorem
(pp. 231-236)
• Appendix III The Möbius Function
(pp. 237-240)
• Appendix IV The Borsuk-Ulam Theorem
(pp. 241-244)
• Appendix V The Galilean Ratios
(pp. 245-248)
• Appendix VI A Candy-Sharing Game
(pp. 249-256)
• Appendix VII Bending Buffonʹs Needle
(pp. 257-262)
• Appendix VIII On Separating Dots
(pp. 263-268)
6. ### Indexes

• Index of Topics
(pp. 269-269)
• Classic Theorems Proved
(pp. 270-270)