Medium | Discrete Maths |

A rectangular table has 100 coins with unit radius, placed on it such that none of the coins overlap, and it is impossible to place any more coins on the table without causing an overlap. Using this specific configuration, find a special configuration of 400 coins which covers the table with overlaps.

Covering means for every point on table there is a coin above it.

Covering means for every point on table there is a coin above it.

Hint

Create coins of radius 2 from the center of all coins. Notice that these coins fill up entire table, they are just bigger than what we are given.

Solution

Consider just one of these coins, with center P. It follows that the center Q of any other coin cannot lie within the coin of radius 2 with center P because it must be at least 2 units away. Thus, we construct all of these coins of radius 2, concurrent with each of the coins of radius 1. If the set of coins of radius 2 did not cover the rectangle entirely, then we could place a coin of radius 1 in this region, contradiction. Thus, the set of coins of radius 2 entirely covers the rectangle.

We now have 100 coins of radius 2 that entirely covers the rectangle. Scale this by a factor of 1/2 in both planar dimensions. Now we have 100 coins of radius 1 that entirely covers a rectangle that is a quadrant of the original rectangle. By placing four of these sets together, we get 400 coins of radius 1 that entirely covers the original rectangle.

We now have 100 coins of radius 2 that entirely covers the rectangle. Scale this by a factor of 1/2 in both planar dimensions. Now we have 100 coins of radius 1 that entirely covers a rectangle that is a quadrant of the original rectangle. By placing four of these sets together, we get 400 coins of radius 1 that entirely covers the original rectangle.

Source: CSEblog

Enable Like and Comment 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

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