## Six: All Directions, Hexagons and 6-letter Words in Scrabble

“Six” 中文版 6： 六合之内

> A cube has 6 faces.

The 3D space we live has 6 directions: up, down, left, right, front and back.

> 6 is an even number. And it is a complete number.

Note: for any number, we can add all of its proper divisors (except itself), and find the sum. If the sum of all its proper divisors equal to the number, then we call this number as a complete number.

For 6: it has four divisors: 1, 2, 3, 6. We strike out 6 (as it is not proper). Then we find that:

1 + 2 + 3 = 6,

which leads us to the conclusion that 6 is a complete number.

The next complete number is 28.

6 also relates to the integer solution of the following equation:

x + y + z = (x) (y) (z)

Solution to this question, with no particular order, is 1, 2, and 3. Which makes both hands equalling 6.

> For any number ending in “6”, when you square it, the last digit of the square is also 6.

For example: 16^{2} = 256, 36^{2} = 1296.

> We have the following interesting identities:

(1 ⁄ 2) – (1 ⁄ 3) = 1 ⁄ 6

and

(1 ⁄ 2) + (1 ⁄ 3) + (1 ⁄ 6) = 1

> 6 has a very interesting rational expression:

6 = (17

^{3}+ 37^{3}) ⁄ (21)^{3}

> A hexagon has 6 vertices and 6 sides. If we connect the alternate vertices, we get an equilateral. This equilateral is similar to the equilateral when we divide hexagons up into 6 smaller equilaterals.

A tetrahedron (4 faces) has 6 edges.

A rectangle has four sides; but adding the two diagonals, there are 6 linking lines connecting each pair of vertices of the rectangle.

> 6 is also a triangle number (defined as follows):

1 = 1

3 = 1 + 2

6 = 1 + 2 + 3

.. .. .. ..

15 = 1 + 2 + 3 + 4 + 5

.. .. .. ..

36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8

All numbers appearing at the left-hand side are triangle numbers.

It is interesting to note that 6 and 6 x 6 = 36 both appear (respectively as 3^{rd} and 8^{th}) in this sequence.

> A regular hexagon can be inscribed to a perfect circle, with the side being equal to the radius. With radius being 1, the circumference of the circle is greater than 6: the perimeter of the hexagon. We know that *π* (for circles) is defined as the ratio of circumference to the diameter. This definition, combined with the fact that the circumference of the circle is greater than 6, has established that *π* > 3: one of the earliest fact about circles for humankind exploring the shape of perfect circles.

> If you know many 6-letter English words, you may score a high point in scrabble (a game for guessing English words). Pause to think of 6-letter words, your mind may generate some 5-letter or 7-letter words first (or some times 3-letter, 4-letter words) before writing any 6-letter.

There are in fact lots of 6-letter words, such as:

artist, sister, attire, entire, entree, orient, season, desert, tested, trophy

In mathematical terms, here are some:

>> The English word **“number”** has 6 letters. **“Figure”** also has 6 letters.

There are some 6-letter English words that you use a lot in the exam: e.g. “answer” and “choice”. When you deal with 3D solids, you may be asked to calculate the “volume” – there, another one.

(But English words *“shape”, “angle” “graph” “ratio”, “speed”, “digit”* each has 5 letters only. You may write “shapes”, “angles”, “graphs”, etc. but this is somewhat cheating!

“*Fraction”* and *“decimal”* has respectively 8 and 7 letters.)

A list of compiled 6-letter words (one word per line) may fill over 100 pages, as using double space format. That’s a lot!