Dots and planes
Five dots are arranged in space so that no more than three at a time can have a single flat surface pass through them.
If each group of three dots has a flat surface pass through it and extend an infinite distance in every direction,
What is the maximum number of different lines at which these surfaces may intersect one another?
If each group of three dots has a flat surface pass through it and extend an infinite distance in every direction,
What is the maximum number of different lines at which these surfaces may intersect one another?
Labels: friday special, puzzle, thinktank
posted by Rajesh Lal at
9:33 AM
14 Comments:
the answer to this question: 45 lines max.
I believe the answer is lucky number 13.
4 dots can be arranged in a tetrahedron to produce 6 lines. These are the edges of the tetrahedron. Lets say the vertices are ABCD
If you add another dot outside the tetrahedron called X, you add four lines connecting from each of the four corners of the tetrahedron to X. These are formed by planes created by pairs of vertices and X. E.g. ABX plane and BCX plane form the line BX. Note these pairs all have a common vertex other than X.
In addition, 3 more lines are added from pairs that don't have a common vertex other than X. E.g. ABX and CDX, ACX and BDX, ADX and BCX. These lines run through X, but not through any of the other vertices.
It's a bit hard to visualize and helps with a drawing.
you are all crazy! the answer could go on forever because of all the dimensions you could put them in.
i agree, are the solutions ever posted or are we meant to choose which is the most logical from the posts?
The first thing we need to know is how many planes are there.
If there are 5 dots from which we are selecting 3 to make each plane we need to know how many distinct combinations of 3 there are in 5. We can calculate this value like this (5 x 4 x 3) / (3 x 2 x 1) = 10.
So knowing that we have ten planes we now need to work out how many of them intersect each other. Given that there are only 5 dots and each plane is made up from 3 dots we can see that any two planes must share at least one dot. Any planes sharing a dot that extend in every direction must intersect, therefore all 10 planes intersect every other plane.
The first plane intersects 9 others the second 8 others (plus the first which we have already counted) the next 7 others (plus the first two) and so on. so the answer is 9+8+7+6+5+4+3+2+1 = 45. (as aaryan said)
The problem with Teo & aaryan solution is that there are Intersect lines that are counted more than once.
For example : 5 Dots = A,B,C,D & E; Then, Plane ABC, ABD & ABE will have only 1 intersect line AB.
My answer = 25 Lines
As long as we remain in three dimensions, there are 9 possible planes (not 10 like teo said). Five dots: A, B, C, D, and E. Three dots make up each plane, so the 9 planes are:
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
CDE
Now it's a simple matter of counting the intersections:
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
The 9 planes have 10 intersections. It doesn't matter that the plane's sizes are infinite; once they intersect they will never intersect again.
Sorry Johnny, but there ARE 10.
You have missed out BDE.
How did I miss that? I spent an hour looking at your equation trying to figure out why it didn't give (what I thought to be) the correct result. Guess I should've double-checked my work first. :-( So 10 planes, 10 intersections.
Well Johnny, how about intersection of:
- ABC & CDE
- ABC & BDE
- ABC & ADE
- Etc...
These intersection lines will not be along 2 of 5 Dots (Eg.:AB,BC,..,DE; a total of 10 intersect lines).
These 10 intersect lines are counted 3 times by Ted's formula; So, I come up with 25 Lines.
(45 - (2x10))=25.
The answer is one.
--"Five dots are arranged in space so that no more than three at a time can have a single flat surface pass through them.
If each group of three dots has a flat surface pass through it and extend an infinite distance in every direction,
What is the maximum number of different lines at which these surfaces may intersect one another?"--
~~~~~~~~~~~~~~~~~~~~~~~~~~~
If each group of three dots has a flat surface (plane) pass through them, and there are only five dots, the answer is ten (10).
First, we will place three dots on a plane. Lines between them will form a triangle (3 lines). Then you place one dot above the triangle, and one dot below the triangle. Group the upper dot with two of the dots comprising the triangle, the connect with lines. Do this and create a plane between each point of the first triangle, you create three additional triangles, but only three (3) more lines.
Do the same with the dot below the first plane, and you create three more triangles, but only three more (3) lines.
A visual representation of these lines would look like a triangle from above, but a diamond when viewed from the side.
Counting the total from the information shown you will find only nine (9) lines, plus one more running from the upper dot to the lower dot.
A grand total of ten (10) lines will be found.
it seems that the question keeps getting asked, but no one ever answers it...
is there a place that we can look to get the "correct answers"..well what the people who write them think the answer is..
i tried google'ing the questions, but alot gets lost in the translation ..know what i mean..
so is ther a place that we can find the answers?
An amendment to my first answer.
The number of unique lines created by planes sharing two vertices is C(5,2) = 10
The number of unique lines created by planes sharing exactly one vertex is 15. 5 (for each vertex) * C(4,2) / 2
There are no pairs of planes sharing zero vertices since there are only 5 dots.
So the total number of unique lines is 10+15 = 25.
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