**Professor Simonson History of Math Ingenuity**

**Fall 1997 MA 149
**

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.