Solutions
... For modulo 2, however, there are only 2 = 4 possiblities: (even, even), (even, odd), (odd, even), and (odd, odd), so there will be congruent pairs of points. For the least number of such pairs, we can make 3 points of one type, such as (even, even), and 2 points in each of the other types. Since the ...
... For modulo 2, however, there are only 2 = 4 possiblities: (even, even), (even, odd), (odd, even), and (odd, odd), so there will be congruent pairs of points. For the least number of such pairs, we can make 3 points of one type, such as (even, even), and 2 points in each of the other types. Since the ...
![arXiv:1003.5939v1 [math.CO] 30 Mar 2010](http://s1.studyres.com/store/data/016290525_1-37241e0159a202e5bf035fd4e5a48270-300x300.png)
Comparing and Ordering Integers
... Two numbers are opposites if they are the same distance from 0 on a number line but are on opposite sides of 0. For example, 3 is the opposite of 3. The opposite of 0 is 0. ...
... Two numbers are opposites if they are the same distance from 0 on a number line but are on opposite sides of 0. For example, 3 is the opposite of 3. The opposite of 0 is 0. ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.