Saturday, June 13, 2015

Writing Roman Numerals

How did Roman Numerals begin?

Many people believe Roman Numerals began as a tally or marking system used by shepherds to keep track of how many sheep they had.
Each sheep was counted with a single notch cut into a stick with a knife. Every fifth sheep was recorded with two notches to form a V and then each tenth sheep was denoted by an X.
This method of record keeping was still being used by Italian shepherds in the nineteenth century.
The Basics

I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1000

The mnemonic 'I Value Xylophones Like Cows Do Milk' may really help you remember (Thanks to Adrian Bruce).

Put It Together

To write larger numbers in Roman numerals, Romans simply "added" all the symbols together. For instance:

LXXIII = 50 + 10 + 10 + 1 + 1 + 1 = 73
MDXII = 1000 + 500 + 10 + 1 + 1 = 1512

- When writing roman numerals never use more than 3 of any symbol.
- When you have numerals on the right hand side of bigger ones, ADD.

Try these problems out (answers on the bottom)

1. Convert XXVIII to our number system
2. Convert XXXVI to our number system
3. What is LXII in our number system?

When you have to Subtract

Sometimes you cannot follow the hint of not using more that 3 of any symbol, for instance in writing 9 in Roman numerals. Nine would be written VIIII, but that would break the rule of not using more than three I's. In these cases you need subtract the value of the smaller number from the larger.

IX = 10 - 1 = 9

-When smaller numerals are on the left of larger ones... subtract.

Try these (answers on the bottom again):

1. IV =
2. VL =
3. CM =
4. XC =
5. CD =

Notable Roman Numeral Quote

'Class, please! If you don't learn roman numerals, you'll never know the date certain motion pictures were copyrighted.'
Edna Krabappel (from The Simpsons)

(Highlight the yellow answers to see clearly)Answers: 5. 2016 4. 622 3. 62 2. 36 1. 28
(Highlight again) 5. 400 4. 90 3.900 2. 45 1. 4

You can see a very good explanation of writing Roman numerals written and illustrated by  Adrian Bruce if you  Click Here .

Friday, June 12, 2015

Facts About Pi

The record for calculating pi, as of 2010, is to 5 trillion digits.

If you were to print 1 billion decimal values of pi in ordinary font it would stretch from New York City to Kansas.

3.14 backwards looks like PIE. Look through the back of the paper!

The first million decimal places of pi consist of 99,959 zeros, 99,758 ones, 100,026 twos, 100,229 threes, 100,230 fours, 100,359 fives, 99,548 sixes, 99,800 sevens, 99,985 eights and 100,106 nines. It cannot be proven, but it is pretty definite that the digits of pi are random.

Pi's evolution

Around 2000 B.C., Babylonians established the constant circle ratio as 3 1/8 or 3.125.

The ancient Egyptians arrived at a slightly different value of 3 1/7 or 3.143.

One of the earliest known records of pi was written by an Egyptian scribe named Ahmes (c. 1650 B.C.) on what is now known as the Rhind Papyrus. He was off by less than 1% of the modern approximation of pi (3.141592).

Plato (427-348 B.C.) supposedly obtained for his day a fairly accurate value for pi: 2 + 3 = 3.146.

The father of calculus (meaning "pebble used in counting," from calx or "limestone"), Isaac Newton, calculated pi to at least 16 decimal places.

William Jones (1675-1749) introduced the symbol "π" in the 1706, and it was later popularized by Leonhard Euler (1707-1783) in 1737.

Rear Admiral Grace Hopper


"Grace Hopper and UNIVAC" by Unknown (Smithsonian Institution) - Flickr: Grace Hopper and UNIVAC. Licensed under CC BY 2.0 via Wikimedia Commons -

In 1954 Grace Hopper was the one on the only persons in the world able to program the complier (software) for the UNIVAC computer.

Grace Murray Hopper was born December 9, 1906 and died January 1, 1992. She was an American computer scientist (one of the first in the world) and United States Navy rear admiral. She was one of the first programmers of the Harvard Mark I computer in 1944.

Harvard Mark I
She invented the first compiler for a computer programming She is credited with popularizing the term “debugging" for fixing computer glitches (inspired by an actual moth removed from the computer).
The First Computer Bug From Grace Hopper's Notebook

The Four Color Problem

The Four Color Problem

The four-color problem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. This problem was first stated in 1852 in England.
This problem was solved by Appel and Haken in 1977, who constructed a computer-assisted proof that four colors were sufficient. However, because part of the proof consisted of a long run by a computer, some mathematicians do not accept it.

The magazine Scientific American ran the following as an April Fools joke in 1976. They said that the figure on the left needed 5 colors. They did not publish the four color solution on the right until the next month (May).

Although it's been proven impossible, many amateur mathematicians have tried to come up with a map that needs 5 different colors to color. If someone could have come up with such a map, they would be famous.
See just how hard it is by trying to color the map on the left with 4 colors without looking a the solution on the right.

Wednesday, June 10, 2015

Pencil Puzzles Series #3

You should read Puzzles Series #1 and  #2 before tackling this one.

OK, we've read about Euler paths and what kind of graphs (or figures) can be drawn completely without lifting your pencil. As a puzzle, I've listed 5 different numbered graphs below. Print out the chart and graphs to make a puzzle you can do at home.

Euler Path?
# Vertices
# Even
# Odd





1                                                                                                 23




Tuesday, June 9, 2015

Pencil Puzzles Series #2

You should read the Euler puzzle page, Puzzles Series #1 before reading this page.

Euler proved that you could not cross all seven bridges without doubling back, or in terms of a pencil puzzle, without lifting the pencil from the paper. The seven bridge puzzle looks like this:

When drawn out as vertices and edges. Euler was able to prove that the parity (odd or even) of the vertices is what is important to whether you can draw a figure without lifting the pencil or duplicating an edge. Each vertex (or blue spot) has a number of edges (or lines) going from it. If the number of edges is even, the vertex is called even. If the number of edges is odd, the vertex is called odd. In the above figure, the vertices have 5,3,3 and 3 lines, so all four vertices are odd.

Euler was able to prove that the only graphs (or figures) you can draw without lifting the pencil are graphs with zero or two odd vertices. If there are two odd vertices you must start drawing on one and end on the other.

So which of the figures A,B,C and D can be drawn without lifting the pencil?

You can draw a continuous line that passes through all the vertices every time. This is called a simple path.
If you can draw a continuous line that covers all the edges it is called an Euler path.

Euler and the 7 Bridges Puzzle Series #1

Leonhard Euler (/OY-lər)15 April 1707– 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in many fields. He also introduced much of the modern mathematical terminology and notation. He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.

Euler is considered to be the most famous mathematician of the 18th century and one of the greatest mathematicians to have ever lived. He is also one of the most prolific mathematicians; his books fill 60–80 volumes. He spent most of his adult life in St. Petersburg, Russia and Berlin, Prussia.

The Seven Bridges of Königsberg is a historically notable problem in mathematics. Leonhard solved this problem in 1735.

The problem was to find a way to walk through the city that would cross each of 7 bridges once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side (the walk does not need to start and end at the same spot). Euler didn't show a way to walk the seven bridges, but he proved that the problem has no solution- there was no way to do it.

To solve this problem, Euler realized that the shape of the island and city do not matter. The only thing that matters is the number of bridges and where they connect, like so:
Konigsberg bridges.png7 bridges.svgKönigsberg graph.svg
In modern mathematical terms, each land mass becomes a "vertex" (blue dot), and each bridge becomes an "edge” (line), the resulting mathematical figure is called a “graph”.

If you can draw a line that starts on a vertex and covers each edge once and only once without lifting the pencil, you have solved the problem by proving it is possible. Remember that Euler proved that this graph is impossible, which means you cannot draw the graph without lifting the pencil and starting at a new vertex.

Can you guess how Euler knew? Look for a new blog showing you how to tell whether you can draw a figure without lifting the pencil.

Substitution Codes - Secret Decoder Rings

          If you have read the post on substitution codes, you know that you can code and decode using two strips of paper that have the alphabet on them. You can slide one strip right and left to align the correct letters for the amount of offset in the message, like this:

 Y   Z   A   B   C   D   E   F   G   H   I   J   K    L   M   N   O   P   Q   R   S   T   U   V   W   X

 A   B   C   D   E   F   G   H   I    J   K   L   M   N   O   P   Q   R   S   T   U   V   W   X   Y   Z

The green strip has been moved over two letters, so each coded letter is 2 more in the alphabet. Notice that we had to take the extra letters in the green strip and place them at the beginning. This is what happens automatically if the strips of the letters are on a ring. A ring! We could call it a decoder ring!!!

Little Orphan Annie's Secret Code Pin

The first secret code pins were made by the makers of Ovaltine (a chocolate drink). They were featured on their radio program, Little Orphan Annie, in 1934. The sponsor started a fan club, Radio Orphan Annie's Secret Society. It had a membership pin and a member's handbook. Inside the handbook was a very simple code (see substitution codes) that any one could learn to decipher (decode) and encipher (code) messages: each letter's position in the alphabet was given a number double its numerical position (i.e., A=2, B=4, C=6, ..., Y=50, Z=52).

                                                     Jonny Quest's Secret Code Ring

By the early 1960s, "secret decoder rings" were what people remembered (even if they were pins), so some people developed a few. The best of the lot was the PF Decoder Ring, which was on the Jonny Quest television show. It substituted letters for letters, and even had a secret compartment! Later, Kix cereal produced a Kix Secret Agent Decoder Ring, which was a cereal premium.

                                                     Midway School Secret Code Ring

We used a 3D printer to make up secret decoder rings for the kids in the 3rd grade. This is a design from Thingiverse . If you are interested, click on the name. The ring is called:

Caesar Cipher Decoder Ring Rounded by cymon

Published on December 21, 2011

Creative Commons - Attribution - Share Alike

If you are interested in a decoder ring of your own and live in the Midway School area, drop me an email.

Monday, June 8, 2015

Substitution Codes - Augustus' Code or Caesar's Cipher

I bet you never thought you'd read an ancient Roman history book, but here is what a Roman historian named Seutonius said about a code that the Roman Emperor, Augustus, used for writing secret messages: (translated from the original Latin)

"Whenever he wrote in cipher, he wrote B for A, C for B, and the rest of the letters on the same principle, using AA for X."

(from the book) Augustus, 88AD

According to Seutonius this is the same method Caesar used, but he substituted D for A, E for B, and so on.

So, Augustus' code works like this:
Each letter in the message becomes the next letter in the alphabet. This is called a substitution code because each letter is substituted for another letter, in this case, the next letter in the alphabet.

 becomes          NFFU NF  BU UI F  SJWFS  DSPTTJOH

OK, this is a pretty simple code. Augustus probably used it when he was "campaigning" or away from Rome  fighting a war. Messages would have had to be sent by courier on horseback, probably to other sections of his army very close to enemy territory. He kept it simple so he could write messages in code quickly and his officers could decode them easily. The code was just hard enough to make sure that if the courier got caught, the enemy couldn't decode it (remember that very few people in Augustus' time could even read at all).

You can make Augustus' code a little harder by shifting the alphabets a little farther so that instead of using the next letter in the alphabet (shifting 1 letter), you could maybe, for example, shift 4 letters of the alphabet so that A becomes E like so:

A   B   C   D   E   F  G   H   I   J   K   L  M   N   O   P   Q   R   S   T   U  V  W  X   Y   Z
E   F    G   H   I    J   K   L  M  N O   P   Q    R   S   T   U   V  W  X   Y  Z   A   B  C   D

This is the code that Caesar used. When you shift a different number of letters, the person getting the message to decode has to know how much you shifted. You can handle this in two ways, you can let them know ahead of time how many letters to shift or you can start the coded message with the letter that tells how far to shift. In the above example of a 4 letter shift, you would start the message with an "E" to show a 4 letter shift.
So, with a 4 letter shift the message becomes:

The message    MEET    ME    AT    THE    RIVER      CROS S  I NG
 becomes       EQ I I X    QI     EX     XLI    VMZIV     GVSWWMRK

One other thing that Caesar did. He knew that you can sometimes partially decode a message if you can see the lengths of the words and how they are arranged. For instance, if you see XLI in the above message, you could guess that it was the word THE and you would know the code for T, H and E. So messages where arranged so that they were in groups of 5 letters and it was up to the decoder to figure out the word breaks. The above coded message becomes


so the decoded message becomes:


so the person decoding it would put in the word breaks and come up with


OR MAYBE "MEET MEAT THE RIVER CROSSING" which could cause a lot of problems if there was really a river crossing called Meat and they hadn't been formally introduced.