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Just with these two elements and a series of rules, questions can be discovered, and if we are careful enough, we can use them in our lives.
Situation 1 : Mid-CheckPoint
Here are the rules in this situation :
1) All the dots and lines must be drawn on a euclidean plane. (where all squares drawn on this plane has 4 right angles and all triangles have a sum of 180 degrees for their 3 angles).
2) you can only connect 2 dots with 1 line, after doing that, you must add a point on that line.
3) A dot can only have 3 line segments, maximum
4) No lines can cross, no dots can share the same location
5) A line can connect to the same dot
Question:
how many dots can be added when 8 dots are given at the beginning?
Solution:
In the beginning, you might start plotting 8 dots and try to draw as many steps as you can, but we can see things from the simplest form.
Considering a dot as a single entity with a single property (location, in this case) is not enough...
Let's try to imagine that now a dot as a room, with 3 doors, since all dots can have 3 line segments, maximum.
We know that drawing a line will use up 2 doors, here, the number of doors decreases by 2.
When we add a dot on a line, the line will past through 2 doors and one door will remain unused, here, the number of doors increases by 1. In conclusion, the number of doors will be decreased by 1 for each step.
So in the given scenario, 8 dots are given, so there should be 8 x 3 = 24 doors, since it takes 23 times to subtract 1 to make 24 to become 1 (which is lower than 2 where it is the number of doors required for drawing a line), you can have 23 steps, and one door will remain unused, i.e. one dot will only have 2 line segments.
Situation 2 : the Dot Web
Here are the rules in this situation:
1) all lines and dots should stay on a euclidean plane.
2) 1 line only can connect 2 different dots, but a dot can be connected by several lines.
3) for simplicity's sake, no dots will share the same location.
4) lines can pass through other lines and dots
Question:
How many lines can be drawn if there are 16 points?
Solution :
You might start plotting 16 dots on a plane and start drawing lines. But before counting the lines, we can look for patterns without counting from 1 to a natural number...
Let's start by plotting the dots in a flexible system: plotting dots D;1 to D;n where n is the number of dots we have (here, we use a semicolon to represent the index number, eg: "D;1" means "the dot with the index number 1" or known as "Dot number 1")
for D;n,
we can draw n-1 lines for n-1 dots
( number of dots - the dot itself = n - 1)
for D;(n-1),
we can draw n-1 -1 lines for n-1 -1 dots
(since the line from D;n to D;(n-1) is drawn, we need the second "-1" to subtract)
for D;(n-2),
we can draw n-1 -2 lines for n-1 -2 dots
(since D;n and D;(n-1) are connected to it, we need that "-2")
for D;(n-3),
we can draw n-1 -3 lines for n-1 -3 dots
(since D;n , D;(n-1) and D;(n-2) are connected to it, we need that "-3")
until you reach D;3,
we can draw 2 lines for D;2 and D;1
(since all the dots where D;({x|x>3}) are connected to it already)
and for D;2,
1 line can be drawn
and for D;1,
all dots are connected to it already, so no more lines can be added for D;1
(n-1)+(n-2)+(n-3)+(n-4)+...+4+3+2+1+0, where there are n terms since each term represents the lines added on that dot.
With a little help from Sir Gauss, we know that the sum would be
(0 + (n-1))n / 2 = (n-1)n/2
if there are 16 dots, then there should be
(16-1)16/2 = 120 lines can be drawn.
If you have other dots and lines questions, feel free to share it in the comment section below ~ or a new post could be cool as well ~
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