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integer
(I don't know how difficult this puzzle is.)
Let a(m,k) =
1
Each integer from 1 to 36 is associated with a letter (a, b, or c) as given below.
Place the integer/letter combinations ('ILC's) into the spaces (one ILC per space) of a 6-by-6 grid such that:
*
Posted is a game and, below the game, a question and a question about a related integer sequence.
This is much easier than 'GCD=1 Grid-Path Puzzle #1'.
Consider an n-by-n grid where the integers 1 to n^2 are arranged such that m is in the ceiling(m/n) row and the (m-1)(mod n)+1 column.
Take th
Start with an n-by-n grid. Place n^2 positive integers into the grid such that: Each integer is greater than the integer placed previously. Each integer is left of, right of, below, or above the integ
...tart with an n-by-n grid drawn on paper. (I suggest an n of at least 5 or 6.)
Players take turns writing integers into any of the grid's empty squares. Each integer must have not been used before in...
I call a set P of positive integers primeary if the sum of an odd number of elements of P is always prime.
It's clear what the sum of an odd number k > 1 of integers means; for k = 1, the sum of an
...ent to 1 mod 3. Therefore 10 to the power N is congruent to 1 to the power N, which is 1, for any positive integer N.
Therefore a googol = 10^100 is congruent to 1 mod 3. A googolplex has googol+1 d...
...r one, the musical scale. I heard somewhere that 5-note octives and 12-note octives are natural, for these integers are derived from the continued fraction of (ln(3)/ln(2)). So wouldn't a number syste...
...tion (k+a(k)).
(Position j is where j-o'clock normally is on an unaltered clock.)
Now, each a(k) is an integers such that 1 ...
...grid drawn on paper. (I suggest an n of at least 5.)
Before play each player *secretly* picks a positive integer.
Starting in the upper left square, players take turns filling in one square each m...
The object of this monthly contest is to pick a whole number, an integer, which is as low as you dare go - but the winning number is the lowest number (above zero) which no-one else has also picked! S
... say, out of curiosity, that we form a chain of sequences from a sequence-set, such as the Encyclopedia of Integer Sequences: http://www.research.att.com/~njas/sequences/index.html#L
A chain consist...
...this kind of maze.
I think the maze works and has one solution.
Instructions: At each step you have an integer, n, which can change at each step. The goal is to get to the end of the maze successf...
I noticed one day that 5 squared is 25 which is almost exactly half of 7 squared which is 49. Then I noticed that 7 squared is 49 which is almost exactly half of 100 which is 10 squared. I fount this
...(x = 23/5 MOD 32/5 - which is true, but does it help me?)
So what's the next step? (I'm after the lowest integer x - without trial and...
For any term x in a circular sequence of integers, let's say that x 'chains to' the term located x places after x.
Here are the 'chain graphs' for a couple of permutations of 0123456789 (each connec
...ions can use addition, subtraction, division, multiplication, and concatenation of digits. You may use the integer factorial and square root functions. Parentheses may be used. The digits may appear i...
Here is a problem by J.A.H. Hunter on expressions for successive integers.
Today we have just one '4' and two '7's.' Using these, all three but no other figures, together with any regular mathematic
This seems like something that MUST have been thought about before.
For a positive integer, n, take all of the distinct integers 1 to n. Place them along an integer number-line such that: (Or you ca
...there is a pure-math proof of a solution/non-solution, this is a kind of math-based maze.
For a positive integer n, take an n-by-n grid. Label the rows with the first n primes, and do the same with ...
...square the rows, columns, and main diagonals all total the same value, and the squares are filled with the integers from 1 to m^2. But in this square, the integers in the shaded sub-squares are SUBTRA...
... ?
Here is how it goes :
1 2 3 4 5 6 7 8 9 20 30 22 23 24 25 26 27 28 11 33...
« Smallest available integer which doesn't include any of the digits writing it's rank in the sequence. »
Ex. :...
...re for m = 1 and m = 3 (at which it takes the values 3^2 and 17^2, respectively). Are there other positive integer values of m which make F(m) a perfect square? If not, can you prove your assertion?
...
...ng of digits formed by concatenating the natural numbers in base 10: C = '123456789101112131415...'
Some integers (as consecutive digits) occur 'prematurely' in this string; that is, they occur to t...
...ral false solutions, I have finally given up.
What is the number of permutations of the first m positive integers where
GCD(a(k-1), a(k)) = 1
for all k, 2 ...
... is kind of fun to solve.
Start with a n-by-n grid. Place a zero in one of the squares. Then place every integer from 1 to n^2 -1 into the grid (one integer per square) so that (for m >= 1): If delt...
Begin with a finite set of integers (such as all the positive integers
Prove product of any N consecutive positive integers is divisible by N! ( N factorial).
...hickness (number of layers) at every point?
Obviously rectangles with a side ratio of 1:n, where n is an integer, can be folded into such a square packet, but what about other rectangles?
Assume t...
Let q and r be NONNEGATIVE integers. Let m be a positive integer.
Let:
...ems interesting.
Assume we have an n-by-n-by-n SOLVED Rubiks Cube. On the small squares we write all the integers 1 to (6 *n^2), one distinct integer per small square, such that each face of the cub...
Let a(0,m) = 1 (for every positive integer m).
For n = nonnegative integers, m = positive integers,
let a(n+1,m) =
m * sum a(n,k) binomial(m+n,k+n) (-1)^(k+1) /k.
In ascii-art mode:
a(n+1,m)
...een adjacent sheets after each revolution, what is the width of the resulting roll of paper to the nearest integer?...
1. For what integers n>=1 is it possible to place n queens on an nxn chessboard such that every row, every column and every extended diagonal has exactly one queen? By extended diagonal, I mean a diag
Here is a problem by J.A.H Hunter on expressions for successive integers.
It's the expressions game again today. You have just a '1', a '4', and a '9', and any regular arithmetical signs you may kno
Hi all!
I have a question concerning the lists of expressions for successive integers. Has anyone tried to write a program, that does the search automatically ? I ask this, because I intend to try t
prove or disprove: does there exist a positive integer N such that there exist integers r, s, t where N = r^2 + 1 = s^3 + 1 = t^5 +1. If so find the smallest such N and prove it is minimal.
Here is a problem by J.A.H. Hunter on expressions for successive integers.
We're back to figures today: 4, 6 and 8 - all three, but only one of each. Using these, together with any regular mathemati
4 can be expressed by these 8 sums of positive integers: 4, 1+3, 2+2, 3+1, 1+1+2, 1+2+1, 2+1+1, 1+1+1+1. How would you prove that a positive integer N can be expressed by 2^(N-1) sums of positive inte
Here is a problem by J.A.H. Hunter on expressions for successive integers.
You have four 'sixes,' but no other figures at all. Using those four 'sixes', all four of them each time, and also any regu
...tement 'Any n people in Central Park at the same time all have the same birthday' is true for all positive integer values of n.
Isn't it?...
... SM4 = 1234 ... SM10 = 12345678910 SM11 = 1234567891011 etc.
a) Find n such that n is the least positive integer >= 1000
where 7 is a prime factor of SMn.
b) Find n such that n is the least posi...
...ation, the divide and conquer paradigm, program development methodology, constraint satisfaction problems, integer programming, and specification....
Second differences between successive integers cubed equal successive multiples of 6:
. . . -27 19 -8 -12 7 -1 -6 1 0 0 1 1 6 7 8 12 19 27 18 37 64 . . .
This should be enough insight to prove tha
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