How can i prove this discrete math challenge? - discrete-mathematics

A test with 20 questions was applied to 300 people. We know that 8 questions had at least
100 hits and the rest at least 200 hits. Prove that some student got at least 11 questions right.

OK assume exactly 8 questions got 100 hits and the remaining 12 got exactly 200 hits. That means there were exactly 8 * 100 + 12 * 200 = 800 + 2400 = 3200 hits. If no student had at least 11 questions right, then the most any student could have got right is 10. If 300 students each got 10 answers right, that's just 3000 hits. But we know there were at least 3200 hits. Therefore, it can't be that there's no student with at least 11 hits; some student must have at least that many.

Related

Creating a schedule of games with "Conference" and a randomized "Non-Conference" with varied conference sizes

There would be 30 total games where 12 are non conf. and 18 conf.
The conference are no smaller than 8 and no larger than 16 and can have odd or even numbers.
There will be up to a couple hundred schools to matchup.
If there is an odd number there will be non-conf games mixed into the conference schedule.
I am thinking of making it 31 or 32 weeks and adding a bye or 2 byes so that will not have to happen.
I am using swift.
I have tried creating randomizing all possible matchups and then creating the schedule from there and having troubles. I also have done the basic round-robin equation but it doesn't work for odd number without a bye.

Octave/Matlab: Arranging Space-Time Data in a Matrix

This is a question about coding common practice, not a specific error or other malfunctions.
I have a matrix of values of a variable that changes in space and time. What is the common practice, to use different columns for time or space values?
If there is a definite common practice, in the first place
Update: Here is an example of the data in tabular form. The time vector is much longer than the space vector.
t y(x1) y(x2)
1 100 50
2 100 50
3 100 50
4 99 49
5 99 49
6 99 49
7 98 49
8 98 48
9 98 48
10 97 48
It depends on your goal and ultimately doesn't matter that much. This is more the question of your convenience.
If you do care about the performance, there is a slight difference. Your code achieves maximum cache efficiency when it traverses monotonically increasing memory locations. In Matlab data stored column-wise, therefore processing data column-wise results in maximum cache efficiency. Thus, if you frequently access all the data at certain time layers, store space in columns. If you frequently access all the data at certain spatial points, store time in columns.

Predicting patterns in number sequences

My problem is as follows. As inputs I have sequences of whole numbers, around 200-500 per sequence. Each number in a sequence is marked as good or bad. The first number in each sequence is always good, but whether or not subsequent numbers are still considered good is determined by which numbers came before it. There's a mathematical function which governs how the numbers affect those that come after it but the specifics of this function are unknown. All we know for sure is it starts off accepting every number and then gradually starts rejecting numbers until finally every number is considered bad. Out of every sequence only around 50 numbers will ever be accepted before this happens.
It is possible that the validity of a number is not only determined by which numbers came before it, but also by whether these numbers were themselves considered good or bad.
For example: (good numbers in bold)
4 17 8 47 52 18 13 88 92 55 8 66 76 85 36 ...
92 13 28 12 36 73 82 14 18 10 11 21 33 98 1 ...
Attempting to determine the logic behind the system through guesswork seems like an impossible task. So my question is, can a neural network be trained to predict if a number will be good or bad? If so, approximately how many sequences would be required to train it? (assuming sequences of 200-500 numbers that are 32 bit integers)
Since your data is sequential and there is dependency between numbers, it should be possible to train a recurrent neural network. The recurrent weights take care of the relationship between numbers.
As a general rule of thumb, the more uncorrelated input sequences you have, the better it is. This survey article can help you get started with RNN: https://arxiv.org/abs/1801.01078
This is definitely possible. #salehinejad gives a good answer, but you might want to look for specific RNN's, such as the LSTM!
It's very good for sequence prediction. You just feed the network numbers one by one (sequentially).

I literally don't know what I need to do for this but it needs to be done and evaluated within a couple of days

1 It's an Emergency.
2 For one of my programming units we have to create a 'UCAS' points calculator in Net beans. UCAS sort out all of the university stuff in the UK.
3 So I need to create a GUI that has text boxes and buttons and I need to enter the amount of passes, merits, and distinctions I've got on my course, which will give me a total amount of UCAS points.
4 Passes = 70 points
5 Merit = 80 points
6 Distinctions = 90 points
7 There's 18 units in total, so if I enter I have 4 passes (a pass for 4 units) the box with the total amount of UCAS points needs to go up.
8 I need three boxes, one for passes, merits, and distinctions, and the box for the total amount of UCAS points. It needs a calculate button of course and a 'reset facility'. It says that the current date needs to be seen on the interface.
9 It also says that there should be an option to quit the program at any time.
10 It has to be a **GUI** and I literally don't know what to do and I don't want to fail again.
11 Can someone help me?
12 I literally don't know how to make the buttons work or anything.
I don't even need any help I just need someone to show me exactly what to do because I haven't got a clue.

SCAN and CSCAN algorithm

I am having hard time understanding the working of SCAN and CSCAN algorithm of disk scheduling.I understood FCFS,Closest Cylinder Next but heard that SCAN resembles elevator mechanism and got confused.
My book says that for the incoming order :[10 22 20 2 40 6 38] (while the disk currently at 20) the SCAN moving at the start serves [(20) 20 22 38 40 10 6 2]; this requires moves of [0 2 16 2 30 4 4] cylinders, a total of 58 cylinders.
How does the pattern [(20) 20 22 38 40 10 6 2] came?
Let's know what the SCAN(Elevator) disk-scheduling algorithm says:-
It scans down towards the nearest end and then when it hits the bottom
it scans up servicing the requests that it didn't get going down. If a
request comes in after it has been scanned it will not be serviced
until the process comes back down or moves back up.
So, in your case, the current position of disk is at 20. So, as per the SCAN algorithm, it'll scan towards the nearest end, and just after hitting bottom, it scans up servicing the requests back up.
The order is :-
| |
| * current position | * move back up to upside
|---> nearest disk is this one |
| so it'll move down and so on. |
| as it hit the bottom _______
____
Fig :- Demonstration of SCAN algorithm
So, as per the given data, the order will be [(20) 20 22 38 40 10 6 2];
EDIT :-
The only difference between SCAN and CSCAN is that in CSCAN,
it begins its scan toward the nearest end and works it way all the way
to the end of the system. Once it hits the bottom or top it jumps to
the other end and moves in the same direction,unlike the SCAN which
moves back to upside using the same path.
As per CSCAN,the direction of movement would be same uptil bottom and then it'll reverse the path.
So, as per the given data, the order will be [(20) 20 22 38 40 2 6 10]; Notice the change in last three disk positions.
I hope it is clear. Feel free to ask remaining doubts.