Medium | Discrete Maths |

On a table you have a square made of 4 coins at the corner at distance 1. So, the square is of size 1×1. In a valid move, you can choose any two coin let’s call them mirror and jumper. Now, you move the jumper in a new position which is its mirror image with respect to mirror. That is, imagine that mirror is a centre of a circle and the jumper is on the periphery. You move the jumper to a diagonally opposite point on that circle. With any number of valid moves, can you form a square of size 2×2? If yes, how? If no, why not?

Hint

Invariance

Answer

No!

Solution

Source: Saurabh Joshi's Blog

Enable Like and Comment Medium | General |

What is/are the next term(s) in the sequence:

a) 1, 11, 21, 1211, 111221, ?

b) 10, 11, 12, 13, 14, 20, 22, 101, ?, ?

c) (This a sequence made by only 2 & 1): 2,2,1,1,2,1,2,2,1, ?, ?....

a) 1, 11, 21, 1211, 111221, ?

b) 10, 11, 12, 13, 14, 20, 22, 101, ?, ?

c) (This a sequence made by only 2 & 1): 2,2,1,1,2,1,2,2,1, ?, ?....

Hint

a) base-change, b) count numbers, c) count repetition

Solution

a) 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, …

This is a Look-and-say sequence! To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit.

b) 10, 11, 12, 13, 14, 20, 22, 101, 1010, 1111111111

This is '10' written in different bases, from 10 to 1!

c) This a sequence made by only 2 & 1: 2,2,1,1,2,1,2,2,1, ?, ?....

This is a version of Kolakoski sequence, and is its own run-length encoding. Each symbol occurs in a "run" of either 1 or 2 consecutive terms, and writing down the lengths of these runs gives exactly the same sequence. It is the unique sequence with this property except for the same sequence with extra '1' at start.

This is a Look-and-say sequence! To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit.

b) 10, 11, 12, 13, 14, 20, 22, 101, 1010, 1111111111

This is '10' written in different bases, from 10 to 1!

c) This a sequence made by only 2 & 1: 2,2,1,1,2,1,2,2,1, ?, ?....

This is a version of Kolakoski sequence, and is its own run-length encoding. Each symbol occurs in a "run" of either 1 or 2 consecutive terms, and writing down the lengths of these runs gives exactly the same sequence. It is the unique sequence with this property except for the same sequence with extra '1' at start.

Source: Common

Enable Like and Comment Latest solved Puzzles

Color Switches Weird Sequences Intersecting Pillars Consecutive sums Scaling a Square Difficulty Level

© BRAINSTELLAR |