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CONTENTS
 

Introduction

Fermi's Piano Tuner Problem

How Old is Old?

If the Terrestrial Poles were to Melt...

Sunlight Exerts Pressure

Falling Eastward

What if an Asteroid Hit the Earth

Using a Jeep to Estimate the Energy in Gasoline

How do Police Radars really work?

How "Fast" is the Speed of Light?

How Long is a Light Year?

How Big is a Trillion?

"Seeing" the Earth, Moon, and Sun to Scale

Of Stars and Drops of Water

If I Were to Build a Model of the Cosmos...

A Number Trick

Designing a High Altitude Balloon

Pressure in the Vicinity of a Lunar Astronaut Space Suit due to Outgassing of Coolant Water

Calendar Calculations

Telling Time by the Stars - Sidereal Time

Fields, an Heuristic Approach

The Irrationality of

The Irrationality of

The Number (i)i

Estimating the Temperature of a Flat Plate in Low Earth Orbit

Proving that (p)1/n is Irrational when p is a Prime and n>1

The Transcendentality of

Ideal Gases under Constant Volume, Constant Pressure, Constant Temperature and Adiabatic Conditions

Maxwell's Equations: The Vector and Scalar Potentials

A Possible Scalar Term Describing Energy Density in the Gravitational Field

A Proposed Relativistic, Thermodynamic Four-Vector

Motivational Argument for the Expression-eix=cosx+isinx

Another Motivational Argument for the Expression-eix=cosx+isinx
Calculating the Energy from Sunlight over a 12 hour period
Calculating the Energy from Sunlight over actual full day
Perfect Numbers-A Case Study
Gravitation Inside a Uniform Hollow Sphere
Further note on Gravitation Inside a Uniform Hollow Sphere
Pythagorean Triples
Black Holes and Point Set Topology
Additional Notes on Black Holes and Point Set Topology
Field Equations and Equations of Motion (General Relativity)
The observer in modern physics
A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File
On Expansion of the Universe - PDF File
 

A Number Trick

Select a three digit number in which the first and the last digit differ by at least two. Construct a second number by reversing the order of digits in the first. Form a third number by taking the difference of the first two. Reverse the order of digits in the third number to construct a fourth, and add the third and fourth number. The result is the number 1089.

Example: Select the number 732. Note: 7 - 2 = 5 > 2, making this number a valid choice. Next, construct the number 237 by reversing the order of digits. Take the difference:

732 - 237 = 495

From 495, construct the number 594, and add:

495 + 594 = 1089.

Proof: Let the originally selected number be represented by

100a + 10b + c

1.

where a, b, c, are integers between 0 and 9 inclusive, and occupy the hundreds, tens, and units places, respectively. We may assume, without loss of generality, that a > c. Now, construct a new number by reversing the order of digits

100c + 10b + a

2.

Take the difference

100(a - c) + (c - a)

3.

and simplify to

99(a - c) = 99alpha

4.

where alpha = a - c. We would like to represent this number, algebraically, as a number in base ten; i.e., with a form similar to that shown in 1. We begin by rewriting 99alpha as

99alpha = (10 . 9alpha) + (1 . 9alpha)

5.

Since 2 less than or equal to alpha less than or equal to 9 by hypothesis, we know that 18 less than or equal to 9alpha less than or equal to 81 so that we may write

9alpha = 10mu + v

6.

where mu and v are integers between 0 and 9 inclusive, and occupy the tens and units places, respectively. Thus, the Right Hand Side of eq. 5 may be rewritten as

(10 . 9alpha) + (1 . 9alpha) = [10 . (10mu + v)] + [1 . (10mu + v)]

= 100mu + 10(mu + v) + v

7.

If the integer mu + v does not exceed a single digit, then the number,

100mu + 10(mu + v) + v

is the number sought. Let us find all possible values of mu + v for 2 less than or equal to alpha less than or equal to 9:

alpha
9alpha
mu + v
2
18
9
3
27
9
4
36
9
5
45
9
6
54
9
7
63
9
8
72
9
9
81
9

Thus, mu + v does not exceed 9 (in fact, it equals 9 in all cases). We conclude that the number, 100mu + 10(mu + v) + v, is the one we sought, and correctly represents the difference, 99alpha, as a number in base ten, with mu occupying the hundreds place; (mu + v), the tens place; and v, the units place. The number obtained by reversing the order of its digits is then

100v + 10(mu + v) + mu

8.

and the sum of these last two numbers is

100(mu + v) + 20(mu + v) + (mu + v) = 121(mu + v)

9.

But, for all values of a considered, (mu + v)= 9, so that 121(mu + v) = 1089.


Please send suggestions/corrections to:
Web Related: David.Mazza@grc.nasa.gov
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Responsible NASA Official: Theresa.M.Scott (Acting)