262
votes
Accepted
107
votes
Is it possible to place 26 points inside a rectangle that is 20 cm by 15 cm so that the distance between every pair of points is greater than 5 cm?
No, it is not. If we assume that $P_1,P_2,\ldots,P_{26}$ are $26$ distinct points inside the given rectangle, such that $d(P_i,P_j)\geq 5\,cm$ for any $i\neq j$, we may consider $\Gamma_1,\Gamma_2,\...
- 355k
103
votes
Is it possible to place 26 points inside a rectangle that is 20 cm by 15 cm so that the distance between every pair of points is greater than 5 cm?
Jack D'Aurizio's answer is nice, but I think the following is probably the solution intended by whoever posed the puzzle:
Note that $26=5^2+1$. So perhaps we can divide our $20\times15$ rectangle ...
- 4,975
39
votes
Accepted
A goat is tied to the corner of a shed
In business and the trades, at least before everything went to decimal notation for fractions, you would almost never see someone write a number as (for example) $\frac 52.$ Instead they would write $...
- 100k
22
votes
Accepted
Radius of a circle touching a rectangle both of which are inside a square
It is just using the pythagorean theorem:
$a=10$ $cm$
$b=20$ $cm$
$(r-a)^2+(r-b)^2=r^2$
$(r-10)^2+(r-20)^2=r^2$
$r^2+100-20r+r^2+400-40r=r^2$
$r^2-60r+500=0$
$r=50$ $cm$
$r=10$ $cm$
The $r=50$ $cm$ is ...
- 8,943
15
votes
Is there a simple perfect squaring of a 1366 by 768 rectangle?
I looked through some of the data on Squaring.Net (actually, copies at the Internet Archive, since the site is currently down), and there is no $1366\times768$ simple perfect squared rectangle of ...
- 4,368
10
votes
Accepted
Cutting a cake into $2$ pieces of equal area
You just need to find the center of each rectangle.
- 8,943
10
votes
Accepted
Venn diagram with rectangles for $n > 3$
Solving the given problem I tried different approaches, obtaining better and better bounds. Also the last bound is the best, I leave the other too, because they can help to solve similar problems.
...
- 93.3k
10
votes
Partition the remaining rectangle into equal parts.
Observe that any line through the center of a rectangle bisects its area.
Then the line joining the centers of two rectangles partitions the remaining areas equally.
- 10.6k
9
votes
Accepted
How to show that any rectangle in ellipse must be oriented parallel to axes?
Without loss of generality $a>b$. Take an inscribed rectangle with two sides gradient $k\ne0$ and two sides gradient $-\frac{1}{k}\ne0$. Shrink along the $x$-axis by a factor $\frac{b}{a}$. The two ...
- 18.5k
8
votes
Accepted
A combinatorial interpretation of a counting problen
Each combination of rectangle and contained cell determines and is completely determined by a pair of ordered triples: the first lists the rows of the top edge of the rectangle, the cell, and the ...
- 620k
8
votes
A goat is tied to the corner of a shed
As far as I can tell you’re answer is fine and the textbook is wrong. Maybe the misprint was $709/4=177 +1/4$. So the answers are typed almost the same.
- 3,247
8
votes
Accepted
Find the length of AB in the given question.
From similar triangles BFG and BFO, $\angle$BFG = $\angle$BOF = $2\alpha$. Note that the midpoint E is the circumcenter of the right triangle ABF. So, the triangles AEF and BEF are isosceles and $\...
- 99.7k
7
votes
Can the squares with side $1/n$ be packed into a $1 \times \zeta(2)$ rectangle?
This answer answers the question only approximately, but at least does so rigorously.
Theorem 4 from
Moon, J.W.; Moser, L., Some packing and covering theorems, Colloq. Math. 17, 103-110 (1967). ...
- 738
7
votes
Accepted
Deskew and rotate a photographed rectangular image (aka "perspective correction")
1) Only four points (like four corners of a sheet of paper) are enough to do the deskewing, the transform is known as a "homography" (as stated in another answer). See also this answer about these ...
- 1,561
7
votes
Accepted
Find rectangle vertices from 4 points located at rectangle faces.
There are many such rectangles.
Let $La$ be any line through $Pa$. Let $Lb$ and $Ld$ be the lines through $Pb$ and $Pd$ perpendicular to $La$. Then let $Lc$ be the line through $Pc$ parallel to $La$. ...
- 97.7k
7
votes
"Cuboid" not the correct name for 3d rectangle?
The word "cuboid" is not consistently defined across all of mathematics—indeed, there are a lot of terms in mathematics which are not consistently defined. A good rule-of-thumb is that the ...
- 30.2k
7
votes
Accepted
Proof that you can approximate any continuous function using rectangles/step functions within a small error
Let $f$ be a continuous function defined on $[0,1]$. Since $[0,1]$ is compact, $f$ is uniformly continuous. Let $\varepsilon>0$ be given. Then there exists $\delta>0$ such that $|f(x)-f(y)|<\...
- 16.7k
6
votes
Accepted
Interesting rectangles
Japanese tatami mats have an aspect ratio of 2:1. This isn't a very interesting ratio, admittedly. But the many ways that tatami mats can be arranged in a room leads to some non-trivial combinatorics ...
- 43.7k
6
votes
Probability of point inside of rectangle $(0,0),(2,0),(2,1),(0,1)$ closer to $(0,0)$ than $(3,1)$.
Visual hint, this is an exercise that can be solved by basic geometry:
- 871
6
votes
Why do we have circles for ellipses, squares for rectangles but nothing for triangles?
This is a question about linguistics and psychology and teaching, not really about mathematics.
We have special words for things we refer to often. Circles come up way more often than ellipses so it'...
- 97.7k
6
votes
Accepted
How many squares in a rectangle?
$$f(a,b) = \begin{cases}1 & a= b\\b/a & (\gcd(a,b) = a) ∧ (a\neq b)\\ a/b & (\gcd(a,b) = b) ∧ (a\neq b) \\ f(b,a) & b>a\\ \frac{(a-(a \mod b))}{b} + f(b,(a\mod b)) &a>b\end{...
- 1,665
6
votes
Accepted
Given distances from a point to three vertices of a rectangle, find the distance to the fourth vertex
Hint: draw lines through $P$ parallel to the sides of rectangle. Use Pythagoras theorem for the lines $PA, PB, PC, PD$ to make up the system of four equations. Then you get the answer $1$.
Details:
$$...
- 31.6k
6
votes
Venn diagram with rectangles for $n > 3$
Suppose you have a Venn diagram with $n$ axis-aligned rectangles of the same width and height. By a scaling of the axes, we may assume the rectangles are all squares. Let's show that it's not possible ...
- 80.2k
6
votes
How to find area of rectangle inscribed in ellipse.
We're lucky that the ellipse is centered at the origin. :)
In this case, the inscribed rectangle is also centered at the origin. If $P = (x, y)$ is the vertex of the inscribed rectangle at the first ...
- 523
6
votes
Accepted
Find the number of points inside a rectangle if the rectangle is divided into $210$ triangles.
Consider a polyhedron that has 1 rectangle face glued to the "rectangle split into triangles" of your problem. Let $T = 210$ be the number of triangles. We have $1$ rectangular face and $T$ ...
- 3,652
6
votes
Covering a circle using rectangles
By using a hexagonal pattern but offsetting the corners, we can slightly increase the area. If the short sides of the rectangles are all $x<1$, we have:
$$A = 3x\sqrt{4-x^{2}}-\frac{3\sqrt{3}}{2}x^...
- 6,123
Only top scored, non community-wiki answers of a minimum length are eligible