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Questions tagged [blackboard]

In a 'blackboard puzzle' a set of numbers is transformed according to certain rules.

7
votes
1answer
221 views

Positive integers on a blackboard

Consider a blackboard with some positive integers written on it. A move consists one of the following actions: Choose two integers $m$ and $n$ on the board, remove them, and write $m+n$ on the ...
4
votes
1answer
112 views

Aatif averages numbers on the blackboard

Aatif has averaged numbers and made the final number $2$: Averaging numbers on the blackboard Today Aatif once again sees the numbers $ 1 , 2 , 3 , .... , 2016 $ written on the blackboard. In one ...
12
votes
2answers
2k views

Averaging numbers on the blackboard

Aatif sees the numbers $ 1 , 2 , 3 , .... , 2016 $ written on the blackboard. In a move Aatif can pick any two numbers on the blackboard, erase them and write instead once their average. As an example,...
11
votes
2answers
321 views

Make a certain number with an unusual calculator

The Kasio Kool-Kalk™ is a poorly-designed calculator with only two buttons, labelled $[A]$ and $[B]$. Pushing $[A]$ replaces the currently displayed number $x$ with $2-\frac{1}{x}$. Pushing $[B]$ ...
11
votes
3answers
325 views

Blackboard problem with polynomial

At the beginning the blackboard contains $n$ real numbers, one of which is $0$. In every step, we may take any polynomial such that all its coefficients are currently on the blackboard, we compute all ...
8
votes
3answers
297 views

Blackboard problem with 2016

At the beginning the blackboard contains a single integer $N$. If the blackboard contains some number $x$, then we may additionally write the two numbers $\displaystyle\frac x {x+2} $ and $ 2x + 1 $ ...
8
votes
4answers
510 views

Blackboard problem with polynomial

The blackboard contains the two numbers 0 and 120. Each minute you are allowed to write an additional number $ x $ on the blackboard if $ x$ has not been written before on the blackboard and if $ x $ ...
18
votes
2answers
2k views

Blackboard cleaning

The blackboard contains 30 integer numbers from 1..2000. Each minute you are allowed to elect some positive integer number x and subtract x from some (chosen by you) of the numbers on the blackboard ...
5
votes
5answers
404 views

The smallest value on the blackboard

The square numbers $ 1^2, 2^2, 3^2, 4^2, \cdots, 100^2, 101^2 $ are written on the blackboard. Each minute any two numbers are wiped out, and the absolute value of their difference is written instead....
27
votes
3answers
1k views

Numbers on the blackboard: From 2-2015 to 1-2014

The numbers $ 2, \ldots, 2015 $ are written on a blackboard. Each minute any two numbers $ x $ and $ y $ are wiped out and are replaced by two numbers $\displaystyle \frac { 4x + 3y } { 5 } $ and $...
14
votes
1answer
841 views

Triangle of numbers on a blackboard

Ten rows of numbers are written on a blackboard. The first row has one $1$, the second has two $2$'s, and so on up through the tenth row with ten $10$'s: $$ \begin{gather*} 1\\ 2,\;2\\ 3,\;3,\;3\\ \...
13
votes
2answers
2k views

The last number on the blackboard

The numbers $ 1, 2, \ldots, 500 $ are written on a blackboard. Each minute any two numbers are wiped out and their positive difference is written instead. At the end only one number remains. Which ...
7
votes
2answers
484 views

Numbers on a Blackboard

You are given an empty blackboard. Once a minute, you are allowed to do one of the following: Write two $1$'s on the blackboard Erase two copies of a number, $n$, and replace them with $n-1$ and $n+...
40
votes
2answers
4k views

Ten numbers on a blackboard

The numbers 1 through 10 are written on a blackboard. You erase two numbers of your choice, and write their product plus their sum $$a,b \to ab+a+b$$ So, now there are nine numbers on the board. You ...