MthT 430 Chapter 2a Projects

MthT 430 Chapter 2a Projects

In class September 5, 2007

Formulas

·

Prove the formula

1² + 2² + ¼+ n² =

n (n + 1) (2 n + 1)

Geometry

A solid cube with side length n cm is constructed from 1 cm³ blocks.

·: How many blocks are needed to construct the cube?

·: How many blocks are visible from the exterior?

The Tower of Hanoi - Problem 26.

There is a puzzle - the Tower of Hanoi - consisting of three spindles, with n concentric rings of decreasing diameter stacked on the first. A ring at the top of the stack may be moved to another spindle, provided that is not placed on top of a smaller ring. ¼ Prove that the entire stack can of n rings can be moved onto spindle 3 in 2ⁿ - 1 moves, and that this cannot be done in fewer than 2ⁿ - 1 moves.

A Google search for Tower of Hanoi yielded many discussions of the Tower of Hanoi on the Internet:

·: http://www.cut-the-knot.org/recurrence/hanoi.shtml

An Inequality - Problem 19.

·

Prove the inequality (Bernoulli): If h > -1, then

(1 + h)ⁿ ³ 1 + n h.

File translated from T_EX by T_TH, version 3.77.
On 04 Sep 2007, 20:47.