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Previous answers seem to say that there are two solutions: $$x=\frac{1 \pm \sqrt{5}}{2}$$ but for $$x=\frac{1 - \sqrt{5}}{2}$$ the base 10 representations of $x$ and $x^2$ are: $$-.618\dots$$ and $$.3 …

Solution to Variation 2: So at 12:00 this ball will be labeled: All balls are relabeled in a similar fashion so at 12:00 the bag contains balls labeled:

Can you find two numbers composed only of ones which give the same result by addition and multiplication? Of course 1 and 11 are very near, but they will not quite do, because added they make 12, and …

The following trio of puzzles comes from http://skepticsplay.blogspot.com/ My question is: What is the solution to variation 2? The original author of these puzzles described them as being abstract. …

There are 7 birds so there are Since 3 birds fly every day, One possible schedule for the birds (A,B,C,...,G) ensuring every pair of birds flies exactly once is:

For which values of the positive integer $n$ is it possible to divide the first $3n$ positive integers into three groups each of which has the same sum? This puzzle is from a UK Junior Mathematical O …

A grasshopper is jumping on a number line and starts at its home at zero (i.e., the “origin”). Its first jump will be length 1/2, its second jump will be length 1/4, its third jump will be length 1/8 …

Take the positive integers {1, 2, 3, ...} and color them red, green or blue. Is it true that no matter what coloring is chosen, we can always find three distinct numbers x, y and z so that x, y, z an …

Here is a diagram created by overlapping four circles. The overlapping circles create eight regions. Put a number in each of the eight regions, using each of the numbers 1 to 8 exactly once, so that t …

From the 2011 South African Junior Olympiad: Several people line up in single file. A solitary latecomer wishes to join the queue. Prove that it is always possible for them to join the line somewhere …

Grandmaster Lev Alburt plays at least one game of chess a day to keep in shape and not more than 10 games a week to avoid tiring himself out. Is it true that if he plays long enough there will be a s …

You are given the first 20 digits of π: 31415926535897932384. In each move, you can select a contiguous group of 5 digits and increase/decrease them all by the same integer, provided that each resulti …

Can you arrange the positive integers $(1, \ 2, \ 3, \ \cdots)$ into a sequence $$T_1, \ T_2, \ T_3, \ \cdots$$ such that the sequence contains every positive integer exactly once AND if $i, \ j, \ k$ …