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integers
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
...Start 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 i...
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
...or 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 syst...
...ition (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 ...
...players might try to guess, by observing which squares their opponents fill-in, what their opponents secret integers are. (For instance, is their opponent trying to end the game early, attempting to c...
...ndex.html#L
A chain consists of DISTINCT sequences, which are adjacent 'links' if they contain any common integers.
Now, as intended here, I am referring to the finite number of elements which are...
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
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
...e 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 can imagine plac...
... 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 SUBTR...
...ing 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 ...
...eral 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 ...
Here is a puzzle that I made up today that 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
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).
Let q and r be NONNEGATIVE integers. Let m be a positive integer.
Let:
...eems 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 cu...
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)
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
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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