## Search found 431 matches

Mon Jul 25, 2016 7:01 pm UTC
Forum: Mathematics
Replies: 148
Views: 18980

### Re: Finding k

Use Stirling's approximation . After a bit of algebra, you get: k = floor( n - (n-0.9189)/ln(n) - 0.5) That constant 0.9189 is actually ln(2*pi)/2. Of course it is not valid for n=0 or 1, and with n=2 it gives -1. Otherwise it matches with your list, even though Stirling's approximation is not accu...
Mon Jul 25, 2016 3:07 pm UTC
Forum: Mathematics
Replies: 148
Views: 18980

### Finding k such that (n^k) is the biggest number < (n-1)!

Hi, Solve Find k such as (n^k) is the biggest number < (n-1)! n=5 4!=24 k=1 25=5^2> 24 so k=1 n=7 6!=720 k=3 7^3=343 < 720 as n grows is there any way to find quickly k knowing n ? When n is not very large it easy to find k Here are the list of the first few k 0,0,0,1,1,2,3,4,4,5,6,7,7,8,9,10,10,11,...
Mon Jul 25, 2016 2:27 pm UTC
Forum: Mathematics
Replies: 148
Views: 18980

Hi, Let me start with examples : 2!=2=1*2 the difference between the factors 2 ans 1 is = 1 and 1 is =1! or 0! 3!=6=2*3 the difference between the factors 3 ans 2 is = 1 and 1 is =1! or 0! 4!=24=4*6 the difference between the factors 6 ans 4 is = 2 and 2 is =2! 5!=120=10*12 the difference between th...
Thu May 26, 2016 4:10 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

The game will be fascinating and exciting if only the areas of the 12 pieces are known to everyone at the starting of the game. The players are allowed then to cut the definitive form of the piece during the game. So for the piece area = 19 for example player could cut it as wished on his turn after...
Wed May 25, 2016 7:35 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Maybe if reduce the size of the board we could see easily if there is an unbeatable configuration. Player 1 : grid 5x5 to cut in 7 pieces. Player 2 : grid 5x5 to cut in 7 pieces. Board to fill : 7x7-49 As 50>49 one player will win. So draws are impossible. An exhaustive analysis is possible without ...
Wed May 25, 2016 12:04 am UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Here is the configuration of player2 1 89 1 90 90 2 1 1 2 92 3 1 16 17 109 4 1 16 17 126 5 1 12 13 139 6 1 12 13 152 7 1 12 13 165 8 1 10 11 176 9 1 9 10 186 10 1 9 10 196 11 1 1 2 198 12 1 1 2 200 100 100 200 Each time I re-arrange the board by filling the holes (11) ; 9 owned by my opponent et 2 o...
Tue May 24, 2016 9:08 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

The goal of re-cut is finding new configuration of the 12. So forget 10,10,10,...1,1.
Tue May 24, 2016 8:58 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

The puzzle is not solved yet.
As you are allowed to re-arrange the pieces on your turn you could fill the holes (1x1) by the own pieces 1x1 of player 1 yet in the board.
So it needs little bit computation to find the best configuration.
Tue May 24, 2016 8:27 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

There is no way that works (unless player 1 is nice and didn't create the 1x1 pieces by makink make holes in his big piece). The 10er and 9er pieces are too big to place them in holes in the big piece so they have to be distributed in the 96 remaining places. 8*10+2*9 is 98 the second 9er piece has...
Tue May 24, 2016 7:53 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Here is my winning configuration against the configuration suggested First column : turns 2th column : player 1 who goes first 3th column : player 2 4th column : sum of p1 and p2 5th column : filling of the board 14x14=196 1 89 10 99 99 2 1 10 11 110 3 1 10 11 121 4 1 10 11 132 5 1 10 11 143 6 1 10 ...
Tue May 24, 2016 11:09 am UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

The big weakness of having 11 pieces 1x1 is that you "choice" is known to your opponent. The next turn you have no choice than placing somewhere on the board 14x14 a 1x1 piece.
Starting from this situation your opponent will have more opportunity to place his pieces if they are diversified.
Tue May 24, 2016 10:56 am UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

@Cauchy Assume that at the turn 12 you have this situation : - It is the turn of Player 2 - Player 1 has still 1 piece 1x1. Player 1 has played (turn 11). He has the 12th turn remaining - The board 14x14 is not full yet. If you rearrange the board 14x14 you obtain a remaining square empty of mxm rem...
Mon May 23, 2016 11:22 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Many interesting ideas. Thank you. Either you play first or second you could start by cutting 2 large pieces : 4x10 and 4x4. No matter what your opponent play you could ALWAYS place them. The maximal piece is a square of 10x10 assuming that it has holes. As the board is 14x14 you could place 4x10 ei...
Mon May 23, 2016 11:06 am UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Is it interesting to start first?
If it is then what is the configuration giving more chances to start first?
How to determine : first the sructure of your pieces in terms of area? second : which piece form is optimal?
It is very hard puzzle I think and we need to find the best way to approach it.
Sun May 22, 2016 11:40 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

I knew that the choice of who is going first will be hard to explain.
The rule is been set up to balance the game and to avoid the "pie rule".
Sun May 22, 2016 11:37 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Keep in mind that if you start first with 1x1 as minimal area you need to play this piece 1x1. You can not start with another piece. I don't see this anywhere in the original rules. Could you point it put for me? Quote : "Player red has a piece with area = 1 and the player blue has a piece wit...
Sun May 22, 2016 11:19 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Just as an example one of the players could cut a piece like a snake : *--*---*---* >>>>>*---* >>>>>*--- *---* >>>>>* >>>>>*---* a star is representing a square > is empty space Inside the "snake" he owns the pieces to place a domino 1x3 or a trimino 1*2*1 etc... You are free to imagine an...
Sun May 22, 2016 11:00 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Since the players have to place in an alternating fashion and the first player has to start with one of their smallest pieces, does that mean that the players have to play their pieces in size from smallest to largest as well? It sounds like you're making that assumption, but I don't see it anywher...
Sun May 22, 2016 2:25 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

In fact this rule is very important. There are pieces easier to place (like 1x1) for example). If a player tried to use a piece easy to place then he must loose a turn by placing such piece. This allows the second player to "block" any further placement. I introduced this rule to "san...
Sun May 22, 2016 12:43 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Keep in mind that if you start first with 1x1 as minimal area you need to play this piece 1x1. You can not start with another piece. I don't see this anywhere in the original rules. Could you point it put for me? Quote : "Player red has a piece with area = 1 and the player blue has a piece wit...
Sun May 22, 2016 12:25 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Did not do complete analysis but below strategy has no apparent weaknesses: - Cut eleven separate 1x1 holes into your square, that will give you 12 pieces. - Based on the rules you're guaranteed to go first (unless opponent has the same idea). - Place your big piece anywhere. - You can ...
Sat May 21, 2016 11:17 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

### Re: Open puzzle about game

Good quesstion anyway
First : it is highly unlikely that the 2 players choose the same configuration
Second : If it is the case then they have to take a new re-cut or picking randomly the first player as usual on other games.
Third : that is not the core of the puzzle.
Sat May 21, 2016 10:55 pm UTC
Forum: Logic Puzzles
Replies: 36
Views: 7617

Game name : "Fill the board" Goal of the game : the last player to place a piece on the board win. Material : Board : grid 14*14 squares 2 scissors 2 Bristol boards in form of grids 10*10 squares : two different colors (red and blue) Technicalities : all the squares have to be of the same ...
Mon Mar 14, 2016 7:21 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Limit

Hi,

#p=2*3*5*7*11*13*.....*p
r=int(sqrt(#p))
d = the number of primes between p and r
s = the number of primes < r

What is the limit of d/s when p goes to infinity?

I conjecture that d/s will near 0 when p goes to infinity.
I can not prove it.
Fri Mar 11, 2016 7:32 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Re: Subsets and product of primes

If we pick randomly the set P1 (P2 will be the complement to P of P1) we will obtain C=A+B. C has some probability to be prime. For sure the probability will decrease as Card(P) grow. For larger sets we could use simulation to estimate the probability of obtaining C prime. What if instead of picking...
Wed Mar 09, 2016 6:13 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Subsets and product of primes

Hi, Let P be the set of consecutive prime numbers from 2 to p {2,3,5,7,11,.....,p} We split the set in 2 subsets P1 and P2 such as P1 U P2 = P and intersection of P1 and P2 = Empty set. Example P = {2,3,5,7,11,13,17,19} P1={2,5,7,11,13} P2={3,17,19} Or P1={3,7,17,19} P2={2,5,11,13} Let us compute A=...
Sat Mar 05, 2016 5:32 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Re: Algorithm : Finding the biggest prime ever

The step of sieving potential candidate could be very quick. Example : #19=15 mod 23 hence some primes will be removed like 31,307,353,etc... because if we add 31+#19 the result will be divisible by 23. The same holds for 307,353.... So we could apply the algo to the first prime remaining on our lis...
Fri Mar 04, 2016 7:53 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Re: Algorithm : Finding the biggest prime ever

I can use semi-prime numbers. So the list will be short. I will have then to solve a diophantine equation as the large prime < r is 3109 mn<3019 (where m and n are primes > p It needs lot of work. No one could guaranty me that such number added to #p will be prime. Anyway thank you very much.
Fri Mar 04, 2016 7:37 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Re: Algorithm : Finding the biggest prime ever

Thank you. What I have found to be sure that #p+A is for sure prime is that you need to compute #p mod q where q represents all the primes between p and the last prime before int(sqrt(#p)). Example for #19 you need to compute 425 primes which is undoable when p goes very large. Out of 425 primes I s...
Wed Mar 02, 2016 12:41 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Re: Algorithm : Finding the biggest prime ever

Here is my new draft. I`m sorry my previous mistakes. I`m too lazy to write a complete paper to show how to build a biggest prime. My method starting from an example Primorial 7 I compute : #7=2*3*5*7=210 What is the number A to be added to #7 such as #7+A will be prime for sure? Condition 1 : The i...
Mon Feb 29, 2016 8:03 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Re: Algorithm : Finding the biggest prime ever

As far as I know, no algorithm that doesn't involve just directly testing for primality will ever generate primes with 100% certainty - their structure is "too random". Some algos are *good* at generating primes, or can generate numbers with special forms that are easy to test for primali...
Mon Feb 29, 2016 7:31 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

### Re: Algorithm : Finding the biggest prime ever

Someone has found a counter example with #19 and the pair 47,29
I need to rethink the conditions.
Any idea to improve the algo is welcomed.
Mon Feb 29, 2016 2:00 pm UTC
Forum: Mathematics
Replies: 17
Views: 3989

Hi everyone, Here is my discovery. How to build the biggest prime ever? My algorithm illustrated by an example. Start with some primorial #17=2*3*5*7*11*13*17=510510 Compute a=int(sqrt(510510))=714 List the first few primes > 17 19,23,29,31,37,41.... You do not need to list them all up to the last p...
Sun Feb 28, 2016 8:05 pm UTC
Forum: Mathematics
Topic: Are all the sequence numbers prime?
Replies: 4
Views: 1570

### Re: Are all the sequence numbers prime?

I'm having a hard time figuring out what this sequence even *is* - is the a() function just "the primorial of the largest prime less than or equal to the sqrt of the previous sequence value"? Yes. If a() is prime (let us say q) then the primoiral is #q=2*3*5*3.....*q If a() is not prime t...
Sun Feb 28, 2016 12:24 pm UTC
Forum: Mathematics
Topic: Are all the sequence numbers prime?
Replies: 4
Views: 1570

### Are all the sequence numbers prime?

Hi, Here is a sequence of prime numbers: 3,5,11,17,47,257,510767,.... The algorithm used to build such sequence is : We definie Primorial p noted #p=2*3*5*7*....*p Start from U(0)=3 the first odd prime Compute a(1)=int(sqrt(U(0)))+1=2 U(1)=U(0)+a(1) where a(1) is equal to primorial #2 = 2 U(1)=5 Com...
Mon Feb 15, 2016 5:50 pm UTC
Forum: Mathematics
Topic: Combinatorics : Castle, rooms and doors
Replies: 14
Views: 3219

### Re: Combinatorics : Castle, rooms and doors

Thank you.
Your results look like OEIS sequence :

https://oeis.org/search?q=6%2C82%2C2604 ... &go=Search

and we have an explicit formula here

https://oeis.org/A078099
Mon Feb 15, 2016 4:25 pm UTC
Forum: Mathematics
Topic: Combinatorics : Castle, rooms and doors
Replies: 14
Views: 3219

### Re: Combinatorics : Castle, rooms and doors

Finding a general formula for nxn is doable using a trick. What are the configurations for n=1 with 0,1,3,4 doors ? 0 1 1 4 2 6 3 4 4 1 So for any nxn 0 and 4 doors wil be = 1. Finding 1 door is like finiding 3 doors (symmetrical) If we find a general formula for nxn with 0,1,2,3,4 doors then we wil...
Sun Feb 14, 2016 1:57 am UTC
Forum: Mathematics
Topic: Combinatorics : Castle, rooms and doors
Replies: 14
Views: 3219

### Re: Combinatorics : Castle, rooms and doors

For n=2 grid 2x2 there are 82 configurations possible
For n=3 grid 3x3 there are 2604 configurations possible
Is there any way to have a close formula?
Wed Feb 10, 2016 3:45 pm UTC
Forum: Mathematics
Topic: Combinatorics : Castle, rooms and doors
Replies: 14
Views: 3219

### Re: Combinatorics : Castle, rooms and doors

The questions I`m asking for are not easy to solve.
It requires time.
If you have any documentation about similar problems please let me know.
Thank you.
Tue Feb 09, 2016 10:47 pm UTC
Forum: Mathematics
Topic: Combinatorics : Castle, rooms and doors
Replies: 14
Views: 3219

### Re: Combinatorics : Castle, rooms and doors

Is it not homework. It is in fact a part of an abstract game.
Does it seem too hard (maybe) or am I wrong?