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Welcome to /puzzle/, a board dedicated to puzzles and other fun math/logic stuff.
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Density and volume
'Sup, /puzzle/, I'm in need of help from the mathematically inclined.
I want to make iron plates for lifting exercises.
The dimensions are as follows:
>450 mm diameter of the plate
>34 mm diameter of the hole in the middle of the plate
Now, according to the known information, what would be the other dimensions of the plates with varying weight given that the above dimensions cannot be changed.
For a:
>25 kg plate
>20 kg plate
>15 kg plate
>10 kg plate
>5 kg plate
>2.5 kg plate
Thanks in advance.
A herd of cows
Say there is a herd of an odd number of cows with the property that if you take any one cow away, it is possible to divide the remaining heard into two herds with equal numbers of cows and equal weight. Show that all the cows in the original heard have the same weight.
On the Tower of Hanoi
In the starting position, assume that the disks come in two alternating colours. Number the disks 1 to n, with n the largest disk. Assuming you move the disks optimally to solve the traditional puzzle, consider the states between moves as well as the initial and final states. What fraction of these states have does the disk k(k = 1 to (n-1)) rest on a disk of the same colour?
Resistors and polytopes
For each given polytope, let the vertices be connected by 1-ohm resistors along the polytope's edges.
1. What is the resistance between two opposite vertices of a cube?
2. What is the resistance between two opposite vertices of an octahedron?
3. What is the resistance between any two distinct vertices of a tetrahedron?
4. Generalize the answers of the previous answers to the corresponding n-dimensional polytopes.
A sum of roots.
Show that for all natural numbers n, the equation shown holds.
Some interesting axioms
This is is based on some stuff I've been studying in my free time.
The axioms are statements on mysterious object called points and lines, and the mysterious relation of incidence between points and lines.
The axioms are:
Axiom 1: For any two distinct points, there is exactly one line incident with both points.
(Notation for future posts: If X and Y are the names of distinct points, [X][Y] is the name of the common incident line, where the square brackets are dropped if the bracketed point name is a single letter.)
Axiom 2: For any two distinct lines, there is at least one point incident with both lines.
Axiom 3: There exist 4 distinct points such that no line is incident with 3 distinct points out of the 4.
One simple consequence of these axioms is as follows.
Theorem 1: For any two distinct lines, there is at most one line incident with both points.
Proof: Assume the statement is false. Then there are two distinct lines that have more than one distinct common incident point. Consider two such distinct points we call A and B. By Axiom 1, there is a unique line which is incident to both. This contradicts our 'construction' where 2 distinct lines are incident to A and B. Thus, our assumption must have been false, and the theorem holds.
Corollary: For any two distinct points, there is exactly one line incident with both points
(Notation for future posts: If x and y are the names of distinct lines, [x][y] is the name of the common incident point, where the square brackets are dropped if the bracketed line name is a single letter.)
Other consequences of the axioms(proof left to reader)
-There exist 4 distinct lines such that no point is incident with 3 distinct lines out of the 4.
-For any line there is a point not incident to it.
-For any point there is a line not incident to it.
-Every line is incident to at least 3 distinct points.
-Every point is incident to at least 3 distinct lines
-If there exists a line incident to exactly n distinct points, then:
– All lines are incident to exactly n distinct points.
– All points are incident to exactly n distinct lines.
–There are n^2 + n + 1 distinct points altogether.
–There are n^2 + n + 1 distinct lines altogether.
I'll post more on this later.
cubing
Any cubers? My 3x3 pb is 19.89 Math/puzzle websites
Post any good math or puzzle websites here.The Forum Geometricum is a free and open euclidean geometry journal.
http://forumgeom.fau.edu/
Math Pictures
Post any cool math pictures you have here.
On a complicated sum
Show that s is an integer.
Aliens and handedness
This isn't so much of a puzzle as a thought experiment., but I think this fits here.
Imagine that you have made contact with a distant alien civilization. Neither of you have any physical artifacts coming from the other; all you have is a text communication line. Explain the difference between left and right, assuming you have common language knowledge excepting that related to handedness. Go in as much detail as possible.
A particular fractal
A fractal is generated from an equilateral triangle by attaching inverted equilateral triangles to each vertex, scaled down by some factor r. At each step in the fractal generation, you attach triangles to the newest vertices. scaled down by r and inverted from the previous step's triangles. (see picture)
Some values of r lead to gaps between 'branches' of the completed fractal, while others lead to overlap at some finite step. What value of r leads to contact at infinite steps?
A puzzle that came from a test.
This text is modified from the original, but the idea is the same.
A particular company organizes its employees into committees. Each employee can be a member of multiple committees. Several of the committees can be in turn organized by a directory, A committee may be listed in several directories.
The employees, committees, and directories follow these rules.
1. For any pair of employees, they are both in exactly one committee.
2. For any employee and directory, the employee is in exactly one committee listed by the directory.
3.All committees have an odd number of employees, and a committee has 2k+1 employees iff it is listed in k directories.
The company has 2401 employees. How many directories are there?
Did you know that it is required to enter a subject when starting a topic or thread on this board i
can someone explain to me what is going on in the sticky picture, I've been really curious and can't seem to find anything, is it some sort of logic chart or something?
n-polytope edge lengths
Of the 5 platonic solids, the cube, the the octahedron, and the tetrahedron alone have regular analogues in all positive dimensions. (Respectively, the n-(hyper)cube, the n-orthoplex, and the n-simplex.) Also, in their respective space, it is possible to circumscribe these regular polytopes with the boundary of some n-ball. What is the ratio of the edge length over the radius of the ball for these three polytopes in terms of n?
Just a question
What is the probability that the (10^100)^100th digit of pi is 7?
Finding roots geometrically
Show that this procedure for finding the real roots of a real quadratic equation finds all and only the real roots.1.On the x-y plane, given the equation ax^2 + bx + c = 0, a != 0, start with the points S = (0, -1), P = (-b, a-c-1) and the line y=(a-1).
2. Construct the midpoint M of S and P and the circle centered at M passing through P.
3. Mark any points where the circle touches y = (a-1). Draw lines from these points to S. Where the line(s) cross the x-axis, the root is given by the x co-ordinate.
(Picture: Example construction using 4x^2 - 8x + 3 = 0)
Pirates and coins.
A group of 7 pirates find a treasure chest. Opening it up, they find only 2 coins within. The pirates divide the treasure in the traditional pirate way: The captain proposes a distribution of coins and there is a vote on whether to accept it. A proposal is rejected if and only if a majority vote against it. If the proposal is rejected, the captain is killed and the highest ranking pirate still alive becomes captain. Each pirate's preferences are as follows, with strongest factors listed earlier.-Staying alive preferred to being killed
-Being captain preferred to not being captain
-Receiving two coins preferred to receiving one coin and receiving one coin preferred to receiving none
-Killing someone preferred to not killing someone.
Assuming all pirates vote logically, which pirate(s), in terms of relative rank in the group at the start, get the treasure?
Geometry challenges
I found a web app that lets you make compass and straightedge constructions. It also gives you some specific challenges to aim for, but they're not mandatory. The pics show how well I've done the challenges.http://sciencevsmagic.net/geo/
(Note, the origin circle is the circle centered on the left starting point passing through the right starting point.)
Distance ratios
Given two points A and B, and a ratio q > 0, !=1, Construct the set of points S such that for all P in S, PA/PB = q.
Rotating Table Puzzle
A rotating square table contains four pockets, one for each corner. Within each pocket, a shot glass lies either upright or upside down. The pockets are such that the shot glass orientation can only be determined by touch. The goal is have all the shot glasses in the same orientation.One is allowed to flip the shot glasses in the following manner: First the table is rotated so that it is impossible to keep track of any particular pocket and stopped in a random orientation. Then, any two pockets are chosen for examination. These pockets are then examined by hand and shot glasses flipped as desired. After this is complete and the hands are out of the pockets, a bell is rung if and only if the goal is met. This entire procedure constitutes a 'move'.
Is there a procedure that guarantees that the goal is met in a finite number of moves?
Numbers and their properties
This is a topic that always interested me. For instance, Mersenne primes, happy numbers, and so on. Post any unique number properties and give examples, I guess.>Lucas-Carmichael Numbers
Take a number a and split it into its prime factors. An L-C Number is one in which the prime factors, when added to by 1, will multiply to give a+1
The smallest Lucas-Carmichael Number is 399.
399 = 3 /times 7 /times 19
400 = 4 /times 8 /times 20
Find the curve
On the x-y plane, for a particular point (a,b), there exists a plane curve such that if a line is incident with (a, b) and a distinct point on the curve (p, q), the slope of the line = q. Find f(x,y) so that f(x,y) = 0 iff (x,y) is on the plane curve.
A starter puzzle
Given any the ordered integer pair(a,b), define the doubly infinite sequence F_{a,b} as shown in the picture. It may be the case that two distinct pairs (a,b) and (c,d) generate the same sequence with merely shifted indices(e.g. (1,1) and (3,5)). Find an expression containing a, b, c, and d that evaluates to 0 if and only if this happens.Solution:
=:>W9\\m_XD:?WA:YW=?W9Z2Z3YA9:X\=?W9Z4Z5YA9:XX^=?WA9:XXjA9:lW`ZDBCEWdXX^a
This looks like a fun board, so please don't be too offended by my drunken incursion.
However, I would argue that "vedic mathz" and other such shortcuts are directly relevant to the subject matter here.
>https://www.youtube.com/watch?v=kvLjpQ0XGao
>https://www.youtube.com/watch?v=grkWGeqW99c
As examples. Nothing amazing, just some kewl shortcuts for people who's calculator ran out of batteries.
Also, I'm not saying study of mathematics makes one insane, but eh, mathematicians tend to be batshit…
>http://www.ams.org/journals/tran/1969-137-00/S0002-9947-1969-0236393-5/S0002-9947-1969-0236393-5.pdf
So, via these premises, how do we be mathematicians without going completely batshit insane?