that are each continuous and last on a certain time interval. Each event has a weight 
However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:
consider 
the sum of 
over all 
such that 
and 
If there are multiple such events, delete the event with the minimal weight.
such that 
and 
.
be a triangle with 
.
be the reflection of 
across the line 
and 
across the line 
.
and 
intersect at 
.
.
be real numbers such that 
and 
.
.
characters from the text, the characters will be the same.
.
is the number of characters in the alphabet, 
is the number of times the 
t
is the number of letters in the plaintext.
which is equal to 
.
,
as we have shown before. This numerator and denominator gives us 
,
'
.
should ALWAYS be encrypted as the letter 
,
.
,
and the frequency of 
.
.
and the letters of our key transformed into numbers are 
.
we use 
where 
.
the corresponding ciphertext number to 
where 
.
t
.
.
to find the equivalent key letter we take 
.
.
.
.
choices because we can choose any letter of the alphabet. Let us say we chose "R".
choices. Let's say we chose "E". Now for the next question mark we can't chose the letter "R" or "E" so we have 
choices. By now you should see the pattern, we have 
choices. That is equivalent to 
.
is the key. 
.
and 
then 
which is a number we cannot convert to a letter. This problem occurs because of the 'wrapping around' from Z to A. To fix this we can use modular arithmetic. If you don't know what this is try googling it. We will almost always be working in mod 26 because there are 26 letters in the alphabet. Our new equation would be:
Y by PikaPika999
Let 
and 
be positive integers with 
. There are tokens placed initially in the leftmost cell of a horizontal row consisting of cells. A move consists of shifting any token 
, 
, 
, 
, 
, or 
positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of and .
and 
be positive integers with 
. There are tokens placed initially in the leftmost cell of a horizontal row consisting of cells. A move consists of shifting any token 
, 
, 
, 
, 
, or 
positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of and .This post has been edited 1 time. Last edited by EmersonSoriano, Apr 5, 2025, 7:26 PM
Reason: change
Reason: change
