Do you know, every prime number p > 3, satisfies the following:
p^2 = 24*k + 1, for some number k.
Also, every prime p > 30,
p^4 = 240*k + 1, for some number k.
Try proving them. Also, an extension of Eratoshenes sieve method gives the following approximation of ratio of first n numbers to primes in them:
Number of primes ~= n*(1 - 1/2)(1 - 1/3)(1 - 1/5)(1 - 1/7)(1 - 1/11).....(1 - 1/pk) where pk is the largest prime < sqrt(n).
Ratio is thus, approximately, (1 - 1/2)(1 - 1/3)(1 - 1/5)(1 - 1/7)(1 - 1/11).....(1 - 1/pk)
Also, a composite number (n) definitely has a prime factor <= sqrt(n). Do you know? Try proving it.
skip to main |
skip to sidebar
| Has anyone seen the movie "The Kingdom of Heaven"? Besides the thing that the movie is very well made, the character of Salahuddin acted very well in that movie. I liked the character so much that I got tempted to keep the name of this blog after it. |
Labels
- Comedy (1)
- Commentary of the contemporary (3)
- Games (10)
- Interior (8)
- Math (14)
- Movies (5)
- Science and Philosophy (3)
Pages
Counter
Followers
About Me
Sudoku
Here comes Sudoku, the big one. Boards have lot more variety than before. Currently there is no hard mode, will add it soon.
| Get Sudoku: | |
| Solve it: | |
| Difficulty level: |
Ponturu
Ponturu is a japanese puzzle in which there are islands and bridges. One needs to connect islands using bridges. One can use only horizontal or vertical connections. Maximum number of bridges between two islands is two. No bridges should cross-over. The number of bridges on an island is indicated by the number inside the circle, for that island. One should make the resulting one, fully connected (there is a path from any island to any other one).
Currently, the ponturu unique solution is not guaranteed.
| Get Ponturu: | |
| Reset Ponturu: |
Cidouri
Here is the collection of Cidouri puzzles. In these puzzles, one should match all red dots with corresponding green dots. One green can be matched to only one red and vice versa. Also, the dots can be matched only horizontally or vertically. The aim of the game is to match all of them, without any cross-overs.
Currently the uniqueness of Cidouri solutions is not guaranteed :-(
| Start a game: | |
| Puzzled! Get back to original board: |
Free Cell
Play FreeCell, card based game
Though it is card based game, it is not chance-based, and is entirely logical. Initially there are 8 stacks of cards, with 4 slots and 4 destinations (at the top of the board). One needs to arrange cards in the destinations using suit, "min to max card" order. One can use the slots and empty stacks as intermediate locations to keep cards. The cards in stacks, whenever you add them to stacks, should be alternate colors, "max card to min card" order.
| Create a free cell puzzle: | |
| Reset the puzzle: | |
| Undo (one step) the puzzle: |
4 in a row
match 4 of your things in a row (horizontal/vertical/crossed). First one which matches so, wins. This plays reasonably fine, and plays very defensive game.
My advice to people is to play on the standard 7*6 board
| Enter 4inARow width here: | |
| Enter 4inARow height here: | |
Flip and match
Pairing-up hidden objects game. Every square contains one entity, and the entity will have to be paired and then those squares are cleared. The game can give colors, numbers, alphabets etc...
When all squares are cleared, game ends
| Enter dimension of the board: | |
| Enter the level of the game: | |
Minesweeper
To mark something as a mine, Hold down the control key, and click on the cell (left-click). The cell will turn into a tulip.
| Enter mine field length here: | |
| Enter mine field width here: | |
| Enter number of mines here: | |
Number Dumber
Number Dumber is a digit arithmetic puzzle, in which one has 4 digits, and the 4 standard math operators +/-/*/// (last one is division), and a final number to make.
One has to make the final number using the 4 digits, using these 4 math ops. No brackets (grouping) allowed, and expressions are evaluated from left to right, as in
7 - 3 * 5 / 1 is evaluated as, 4 * 5 / 1 20 / 1 20Notice that using 4 4's with math operators to generate numbers from 1 to 100, is kind of like a special application of this.
I have written a solver, that solves the NumberDumber, given the digits and the final number. The solver, as well as the puzzle creator are here. Get some puzzles and try solving them, by not using solver.
The puzzles are reasonably tough (how tough can be digit arithmetic :-)), as in they have a unique solution only.
| Enter the first number: | |
| Enter the second number: | |
| Enter the third number: | |
| Enter the forth number: | |
| Enter the result number: | |
| The result is displayed here: | |
Fifteen puzzle
Fifteen puzzle, a puzzle with jumbled numbers and one empty slot, which can be slid. Put the numbers 1 to 15 in order, by the row, like:
1 2 3 4
5 6 7 8
....
1 2 3 4
5 6 7 8
....
Jumbled pictures
A small picture is shown in a jumbled manner. By clicking on adjacent parts, one can swap them. See, if you can get the original picture, by swapping or not.
There are some issues yet, working on them.
Frustrated? Press, solve me:
Frustrated? Press, solve me:
Maze creator
Here is the maze creator, input the width and height of maze (<= 100), and see the maze created. To be a little challenging at least, try with minimum dimension of 15 * 15 etc...
| Enter maze width here: | |
| Enter maze height here: | |
Blog Archive
-
▼
2010
(35)
-
▼
May
(8)
- I also got some interest in the problem of finding...
- Also, I believe n! is not a square number, for any...
- Ok, I give up. I tried doing 101-200 numbers, but,...
- Do you know, every prime number p > 3, satisfies t...
- Reverted the look back to original one, retro look...
- Look at this:http://www.cut-the-knot.org/arithmeti...
- Looks like finding a representation of 75, using o...
- For the mathematically inclined, here is something...
-
▼
May
(8)
All data and information provided on this site is for informational purposes only. This site makes no representations as to accuracy, completeness, currentness, suitability, or validity of any information on this site and will not be liable for any errors, omissions, or delays in this information or any losses, injuries, or damages arising from its display or use. All information is provided on an as-is basis.
No comments:
Post a Comment