Throwback Fact of the Week - Magic Squares - 11/20/14

A magic square consists of a square grid with n rows and n columns, thus it consists of  n² total boxes. These boxes are all filled with different integers. The sums of the numbers in each of the vertical columns, horizontal rows, and the diagonals are all equal.

A magic square is said to be normal if the integers filling the boxes are the consecutive integers from 1 to n². The sum, S, of the rows (columns and diagonals) for a normal magic square is given by the formula: S = n(n²+1)/2

The normal 4x4 magic square below was created in 1514 by Renaissance artist Albrecht Dürer in his engraving titled Melencolia I. 

This square has many amazing and intriguing properties. The sums of each of the rows, columns, and diagonals is 34 (as expected from a normal 4x4 magic square). However, notice that the bottom two central numbers are 15 and 14, i.e., the year the engraving was made, 1514. The numbers in the bottom left and right corners, 4 and 1, correspond alpha-numerically to the engraver's initials A.D.! 

Additionally, the sum of the four corners (16+13+1+4), the sum of the middle 2x2 square (10+11+6+7), the sum of the middle two entries of the two outside columns and rows (5+9+8+12) (15+14+3+2), are all equal to 34. 

Throwback Fact of the Week - Knight's Tour - 11/06/14

Knight's Tours have fascinated mathematicians for centuries. A Knight's Tour is a set of moves made by the knight on a chessboard, on which the piece visits each square exactly one time. Leonhard Euler was the first to write a mathematical paper analyzing these tours. 

If you are unfamiliar with chess, the knight piece may move in an "L" shape; it can move two squares horizontally and one square vertically, or two squares vertically and one square horizontally. 

A tour is called closed or reentrant if the knight ends its tour on a square that is one move away from the square it started (meaning it could begin the same tour over again). Otherwise it is an open tour. 

Apart from the standard 8x8 square chessboard, Knight's Tours have been studied on boards with varying dimensions as well as on irregular (not square) surfaces. 

Below is an animation of an open Knight's Tour on a 5x5 chessboard. 

The Worldwide Lecture Seminar Series Presents: Alexandru Dimca

CAMBRIDGE -- Local and traveling mathematicians gathered Friday at the Center of Math where the Worldwide Lecture Seminar Series presented Alexandru Dimca, Université Nice Sophia Antipolis, on the fundamental groups of complex algebraic varieties. The event was captured in its entirety and has been made available free to the public.

For more Worldwide Lecture Seminar Series coverage please click here.

The Worldwide Lecture Seminar Series Presents Alexandru Dimca on the fundamental groups of complex algebraic varieties.

Una foto pubblicata da Worldwide Mathematics (@centerofmath) on Nov 11, 2014 at 8:33 PST

Throwback Fact of the Week - Sicherman Dice - 11/06/14

Sicherman dice are not your typical board game dice, but you could use them in almost any board game without changing the outcomes. How can this be? 

The faces on the Sicherman dice (pictured below) are numbered 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8. Sicherman dice are very unique; they are the only pair of 6-sided cubic dice with positive integers that have the same odds as your standard cubic dice. Note, if negative numbers are allowed there are an infinite number of such dice.

Image credit: Grand Illusions (dice can also be purchased from their site)

What does this mean? As you know, with regular dice there is a fixed probability of rolling a given sum (you can roll between 2 and 12) when both dice are thrown. The same holds true for Sicherman dice; you can roll a sum between 2 and 12 and the odds of a particular sum occuring is the same as for a regular pair of dice. 

For example, regular dice have a 1/9 chance of rolling a sum of 5. With Sicherman dice, there is also a 1/9 chance of rolling a sum of 5. 

We mention you could play almost any game with these. The reason for almost is because some games have certain rules when doubles are rolled and these dice have different odds of doubles being rolled. The reader is left with the exercise of calculating the odds of rolling doubles using Sicherman dice!

Throwback Fact of the Week - Vampire Numbers - 10/30/14

Vampire numbers were first defined by Clifford Pickover in 1994. 

A vampire number is a composite whole number with an even number of digits, n. The vampire number has form v = xy where the digits x and y each have n/2 digits. These digits consist only of the digits in the vampire number, but in any order. Note that x and y are not allowed to both end in trailing zeros. 

1260 is the smallest vampire number.

 v = xy

1260 = 21*60

You'll notice that in this case, the vampire number has 4 digits; x and y each have 2 digits, and those digits consist of a re-arrangement of the digits in the vampire number. 

The digits x and y are known as the fangs of the vampire number. Vampire numbers sometimes have multiple pairs of fangs that fit the rules for forming the vampire number. Such as 125460 = 204 × 615 but it also equals 246 × 510.

An even more interesting type of vampire number would be a vampire number where its fangs are also its prime factors. For example, 117067 = 167 × 701, where 167 and 701 are the prime factors of 117067, making this number a prime vampire number! 

Throwback Fact of the Week - Benford's Law - 10/23/14

Benford's Law, or more descriptively, the first-digit law, states that in most number lists and real-life collections of data, (such as death rates, baseball statistics, etc.) 1 occurs in the leftmost leading position roughly 30% of the time. It also asserts that larger digits occur less and less frequently, with 9 occurring a mere 4.6% of the time as the first digit.

The significance of this law lies in the fact that it shows that the most common first digit in an arbitrary source of data is not random; if it were random, every digit would be expected to occur first about 11.1% of the time, a 1 out of 9 probability.

The graph below shows probability that each digit occurs as the leading digit. A given set of numbers satisfies Benford's Law if the leading digit d, where d can be the digits 1 through 9, occurs with probability log₁₀(1+1/d).

The law is named after physicist, Dr. Frank Benford, who stated it in 1938. However, it had been previously pointed out by Simon Newcomb in 1881, when he noticed that the pages of logarithms containing the numbers beginning with 1 were much more worn out than other pages.

The law has been used as a method to detect fraud; for example, an accountant could detect a fraudulent tax document if the occurrence of leading digits does not sync up closely with Benford's Law.