## Friday, March 15, 2024

### Logic+ 05: UNIZOR.COM - Math+ & Problems - Logic

Notes to a video lecture on http://www.unizor.com

Logic+ 05

Problem A

A casino manager, analyzing the results of operations of roulette tables during a period of one day, comes to the following statistics:
there were N1 people who won at least 1 time;
there were N2 people who won at least 2 times;
there were N3 people who won at least 3 times;
etc.
there were Nn people who won at least n times;
and nobody won more than n times.

How many times casino has lost in roulette?

Hint A:
N1 ≥ N2 ≥ N3 ≥...≥ Nn

Casino has lost
N1+N2+N3+...+Nn games.

Problem B

How many rooks can be placed on a chessboard such that no two rooks prevent each other to move along chessboard from one edge to another vertically or horizontally?

Problem C

Squares on a chessboard are enumerated as follows:
1st row: 1, 2,...,8
2nd row: 9, 10,...,16
3rd row: 17, 18,...,24
etc.
8th row: 57, 58,...,64

Eight rooks are placed on a chessboard such that no two rooks prevent each other to move along chessboard from one edge to another vertically or horizontally.

What is the sum of numbers on the squares occupied by these rooks?

Hint:
Each number N on a square can be represented as
N=(R−1)·8+C, where
R is the row number (from 1 to 8) and
C is the column number (also from 1 to 8).

Problem D

There is a cup of coffee and a cup of milk. Assume, both cups contain the same amount of liquid.
A spoon of milk is taken from a milk cup and added to coffee.
Then, after stirring the coffee, a spoon of coffee with milk is taken and added to a milk cup.
Which is greater,
a concentration (in %) of milk in a coffee cup or
a concentration of coffee in a milk cup?

Answer D: They are the same.

Problem E

Given a table with each its cell containing some number.
Any number in this table is equal to an arithmetic average of its neighbors - those numbers that this number shares a cell's border with.
If a number is in the middle of a table, it has 4 neighbors (up, down, left, right).
If a number is at the edge of a table, it has 3 neighbors (for example, for a number near the top border the neighbors are left, right and down).
If a number is in the corner of a table, it has only 2 neighbors (for example, for a number in the bottom right corner the neighbors are up and left).

Prove that all numbers in the table are equal to each other.

Hint E: Consider the neighbors of the smallest or the biggest number.