Number Chart

The nth Root 

2

3

...                ...

n

Number Chart

Exponents

   

      Powers of 5 

	.. etc..
52	1 × 5 × 5	25
51	1 × 5	5
50	1	1
5-1	1 ÷ 5	0.2
5-2	1 ÷ 5 ÷ 5	0.04
	.. etc..

Number Chart

 Exponents of Negative Numbers

 

1 (Odd):	(-1)1 = -1
2 (Even):	(-1)2 = (-1) × (-1) = +1
3 (Odd):	(-1)3 = (-1) × (-1) × (-1) = -1
4 (Even):	(-1)4 = (-1) × (-1) × (-1) × (-1) = +1

PUZZLE

The Dual Cabbage Way

 Using three straight lines, divide the cabbage patch up into six sections with two cabbages in each section.

 

Our Solution

Here is one way: 

PUZZLE

Cut Cube

A solid, four-inch cube of wood is coated with blue paint on all six sides.

Then the cube is cut into smaller one-inch cubes.

These new one-inch cubes will have either three blue sides, two blue sides, one blue side, or no blue sides. How many of each will there be?

 Our Solution

There are 8 with three sides colored, 24 with two sides colored, 24 with one side colored, and 8 with no sides colored.
Here I have done one face, and also shown you that "inside" there are 8 with no paint at all:

PUZZLE

Band Around the Earth

The circumference of the Earth is approximately 40,000 kilometers, and someone 

has just made a metal band that circles the Earth, touching the ground at all locations.

You come along at night, as a practical joke, and add just 10 meters to its length (one hundredth of one kilometer !)

It is now one four-millionth longer, and sits magically just above the ground at all locations

How far has it risen ... could a flea, a rabbit or even a man squeeze underneath it?

Solution

Use the formula Circumference = 2 × pi × Radius

Before: Original Circumference = 2 × pi × R
After: Original Circumference + 10m = 2 × pi × (R + Gap)

Subtracting the two:

10m = 2 × pi × Gap

So, the Gap = 10m / (2 × pi) = 1.6m approximately 

So a man could fit under it easily (though he might bump his head)

Note: Adding 10m to the circumference of ANY circle increases the radius by 
10m / (2 × pi), no matter what the original circumference was.   

PUZZLE

12 Days Of Christmas

According to the traditional song, on the first day of Christmas (25th December), my true love sent to me:

. A partridge in a pair tree 

On the second day of Christmas (26th December), my true love sent to me THREE presents:

. Two turtle doves
. A partridge in a pear tree 

On the third day of Christmas (27th December and so on) my true love sent to me SIX presents:

. Three French hens
. Two turtle doves
. A partridge in a pear tree 

This carries on until the the twelfth day of Christmas, when my true love sends me:

Twelve drummers drumming
Eleven pipers piping
Ten lords a-leaping
Nine ladies dancing
Eight maids a-milking
Seven swans a-swimming
Six geese a-laying
Five gold rings
Four calling birds
Three French hens
Two turtle doves
A partridge in a pear tree 

After the twelve days of Christmas are over, how many presents has my true love sent me altogether?

Solution

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 presents

Which is really interesting when you think there are 365 days in a typical year!  

PUZZLE

Dropping Balls

You have to do an experiment to determine the highest floor on a 100-floor building from which a manufactured snooker ball may be dropped without breaking. 

You are given two identical snooker balls, which you can drop from various floors of
the building, to carry out your experiment. 

If a ball doesn't break after being dropped, it may be reused without suffering any loss of quality. But if both balls break before you have determined the highest floor, then you are an incompetent bungler and your boss is ultimately going to fire you.

What is the least number of times you must drop the snooker balls in order to determine the highest floor?

Our Solution

The answer is: 14

You drop the first snooker ball from the 14th floor. If it breaks, you can then determine the highest floor by dropping the second snooker ball no more than 13 times (drop it from the 1st floor, and if it doesn't break, drop it from the 2nd floor, and if it still doesn't break, drop from the 3rd, etc).

If the first ball survives the drop from the 14th floor, you then drop it from the 27th floor (14 + 13 = 27). If it breaks, you can complete your test in no more than 12 drops with the second ball by dropping it between the floors 15 to 26.

If from the 27th floor the first ball still doesn't break, the next floor to drop it from is the 39th (14 + 13 + 12 = 39). If it breaks, drop the second ball from floors 28 to 38 (max 11 drops).
Repeat the experiment the same way whenever the first ball doesn't break; after the 39th floor should be the 50th floor (14+13+12+11), then the 60th floor (14+13+12+11+10), and so on. If the first ball survives 11 drops, you will be on the 99th floor. In that case, it only takes one more drop to complete the whole test. 

PUZZLE

Double Hearts Ratio

Which area is bigger: the total orange or the total red?

The Solution

Using the illustration below, we can calculate the following areas:

A(square) = 9 x 9 = 81;
A(red) = 28.27 + 12.57 = 40.84;
A(orange) = 81 - 40.84 = 40.16.

Thus, A(red) > A(orange).

PUZZLE

Diophantus

We know very little about the life of the mathematician Diophantus (often known as the 'father of algebra') except that he came from Alexandria and 
he lived around the year 250 AD. 

However, there remains a riddle that describes the spans of Diophantus's life:

"This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life."

In simpler English it says: Diophantus's youth lasted 1/6 of his life. He had the first beard in the next 1/12 of his life. At the end of the following 1/7 of his life Diophantus got married. Five years from then his son was born. His son lived exactly 1/2 of Diophantus's life. Diophantus died 4 years after the death of his son.

How long did Diophantus live?

The Solution

There is an equation to reflect the several ages of Diophantus:

1/6x + 1/12x + 1/7x + 5 + 1/2x + 4 = x
So the solution (x) is 84 years.  

PUZZLE

Chain Reaction

Chains. Make-up. What's the connection? The answer lies in a film. The DIY chain of shops, TEXAS, once made a film about their new eye make-up. They called it "The Texas Chain Store Mascara". I digress.
Below you can see several bits of broken chain. I have been told to join up all the pieces to make a complete necklace using all twenty links.

 

However, it is a very fiddly job, and it takes one minute to cut one link, and two minutes to join it up again. How long will it take me to finish the necklace?

Solution

Just 15 minutes!! That's a surprise, isn't it? I'm sure you're on the very edge of your seat to know how it's done.

I'll tell you anyway.

Separate the two shortest lengths of chain by two cuts. This makes two open links and four single closed ones.
Time taken: 2 minutes. 
Cut open three of the single links. This leaves five open links and five pieces of chain

(4, 4, 3, 3, 1).
Time taken: 3 minutes. 
Use the five open links to join the five pieces of chain (each link joins two ends).
Time taken: 10 minutes. 

TOTAL TIME: 15 MINUTES.

PUZZLE

Blockslide

The diagram below shows a cross-shaped box containing three numbered blocks.

 

The puzzle is to slide the blocks around the box until the numbers read 1,2,3 as you go down.

How do you do it?

Solution

  

  

Geometric Patterns

Number Chart

Geometric Patterns