Professor Simonson History of Math Ingenuity

Fall 1997 MA 149

Assignment 3

Write your solutions to these problems as a group. Detail your attempts and ideas whether you succeed in solving the problem or not. You will always have the opportunity of rewriting an assignment after class discussion.

1. A Mystery from India:

The following diagram appears in the Lilavati, a math manuscript by Bhaskara of India from about 1150 C.E. The book is about arithmetic and is named after Bhaskara's daughter. 2 4 5

0 0 1 1 3

6 2 5

9 1 2 3 7

4 8 5

1 0 0 1 2

4 8 0

1 4 0

a. Can you figure out what arithmetic method Bhaskara is teaching and how it works?

b. If you get part a, can you explain why it works?

2. Number Representation

a. Convert the following decimal numbers to Egyptian, Babylonian and Binary notation:

789, 45, 1000.

b. Convert the following binary numbers to decimal and Babylonian notation: 10010011.101 1110.10001

3. Arithmetic Methods

a. Multiply 129 x 518 using the Egyptian method of successive doubling.

b. Subtract 8889 from 9111 using the Vedic method.

c. Subtract 8889 from 9111 using the Complement method.

4. The Simple Nim Game Revisited: What is the correct strategy in the 1-3 stick Nim game when the rules are changed so that you LOSE if you take the last stick.

5. A New Game of Dots: Consider a game for two players played on a grid of dots m by n. Each player uses a different color, and on his turn connects any two consecutive dots in a row or column by a horizontal or vertical line respectively. The first player to make a square using only his color wins. Analyze this game as best you can and try to make appropriate conjectures and/or proofs.

6. A Lost Tablet

This is a fictional problem based on the famous Babylonian stone tablet fragment called Plimpton 322. (You can read about this tablet in Katz, on reserve). Shown below are 3 columns of 7 lines of inscriptions found in a cave in Turkey. The markings resemble no known language, but they may represent some mathematical markings. It is guessed that probably 1/3 of the tablet was lost. Discover an interpretation of these markings and use it to make an educated guess concerning the markings on the missing 1/3 of the tablet.