12
votes
Accepted
Patrick and Rachel go to a tennis tournament with 7 other couples
First notice the wording of the question: "Patrick asked everyone how many people they played against, and found that each one answered with a different number"
This means that
In ...
- 123k
8
votes
7
votes
6
votes
6
votes
Show there is always a pair at least 16 apart
Perhaps I'm missing something but here is a quick proof
- 123k
6
votes
Accepted
Tiling twelve 5 x 10 rectangles with ten sets of the twelve pentominoes
It is easier to halve the size of the puzzle - using 5 sets of pentominoes to cover 6 of those rectangles. If that can be solved, you can simply use two copies of that as a solution.
I'll restrict ...
- 43.1k
6
votes
Can the Spider catch the Fly?
Assumption (& simplification):
The spider and the fly starts at different finite points on an infinite 1-d axis.
Furthermore, assume the fly starts somewhere. Whether it's 10, 100 or 1000 it ...
- 2,689
5
votes
Elegant solution to this digit puzzle
Overall, I've counted
so there will inevitably be some degree of case bashing.
This is how I would proceed.
Notice firstly that
Suppose $f < 9$
Now suppose $f=9$
- 123k
5
votes
5
votes
5
votes
Accepted
5
votes
4
votes
What size square grid can you tile?
I'll generalise it slightly to ask what $m\times n$ grids can be tiled.
The tile has area $4$, so clearly we need $mn$ to be divisible by $4$.
- 43.1k
4
votes
Accepted
Bertrand's Ballot Theorem
Let $y$ be the current number of votes for $B$, and $x$ be the number of current votes for $A$. If $y>x$ at any time, then $x=y$ at some point in this count. So we have to find the probability that ...
- 56
4
votes
Good Rectangles and Evil Numbers
So it is proved that only the following set might be evil
I wrote an MIQCP model and solved with gurobi for the remaining numbers, proving they are truly unfeasible and so evil.
(My model can only ...
- 486
4
votes
Accepted
A strange corporate
I came up with:
Rationale:
Double checked myself in Excel, and discovered that I should have finished my handwritten version.
And for the bonus question:
- 1,211
4
votes
Elegant solution to this digit puzzle
I just wanted to 'see' what the solutions looked like, but I disqualified myself by using a computer to get the solutions. Anyway, here is a visualization for your amusement.
Visualization
...
- 2,213
3
votes
How high does the ladder reach up the wall?
Let h,w be the height and width of the triangle formed by the ladder with the wall and the ground.
First let us divide through with d: $H:=h/d, W:=w/d, L:=l/d, D:=d/d=1$
Then
$L^2=H^2+W^2$ (1).
...
- 11.6k
3
votes
Accepted
Productive Neighbours
A solution by hand:
Now that it's known which digits fall in each loop, the exact placement is straightforward.
- 1,075
3
votes
A barrel contains 10l of whiskey, another one contains 10l of coca-cola. If I do the following operation
Never mind any of the transfer operations: the amount of whiskey in the cola barrel will be
This is enough to tell us that whenever there's an equal amount of liquid in both barrels, the amount of ...
- 68.2k
3
votes
Accepted
Will Romeo meet Juliet?
Let $R$ and $J$ be uniform $[0,1]$ random variables to represent the arrival times of Romeo and Juliet, respectively. The star-crossed lovers meet if $|R-J|\le 1/4$.
The desired probability is the ...
- 7,429
2
votes
Accepted
How many colleagues went to Starbucks?
If you have $25$% of the total coffee and $50$% of the total milk:
Using this, we find:
So there are:
- 1,954
2
votes
Can the Spider catch the Fly?
Yes, the spider can always catch the fly.
First, observe a simpler case:
Now, let the start distance to be unknown:
- 650
2
votes
Accepted
A bunch of circles and squares
Original Partial Answer (Complete Answer below this one):
Weeks ago I figured out that the gray section "IN" means
thinking that the values in order are
At first I thought the slopes of ...
- 13.1k
1
vote
Productive Neighbours
The unique placement is
where x =
Here is my thinking:
Then I had the idea to brute force it, so I
Then, I named each region in the diagram a letter from A-G and added a few columns to calculate ...
- 13.1k
1
vote
Good Rectangles and Evil Numbers
The evil numbers are
as already stated by @xd y . What this post adds is a simple non computer aided proof:
a) these are really bad:
Given a tile t and an x offset we say t covers x if there is a y ...
- 11.6k
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