Questions tagged [blackboard]

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

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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 ...
xnor's user avatar
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29 votes
2 answers
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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 $...
Haobin's user avatar
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20 votes
2 answers
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The last number on the blackboard

The numbers 1, 2, ..., 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 values can ...
Haobin's user avatar
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19 votes
2 answers
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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 ...
Haobin's user avatar
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16 votes
6 answers
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Abby and Bobby and three numbers on the blackboard

Abby selects a non-negative integer $A$, and Bobby selects a non-negative integer $B$. They then both tell their number secretly to Summy. Summy writes the numbers $5$, $8$, and $15$ on the ...
Gamow's user avatar
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14 votes
1 answer
926 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\\ \...
Julian Rosen's user avatar
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13 votes
2 answers
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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,...
D.A.G.'s user avatar
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13 votes
1 answer
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99 numbers on the blackboard

The numbers 1,2,3, ..., 99 are written on the blackboard. Alice and Bob play the following game, taking turns. Alice starts. At every turn, each player removes one number until two numbers are ...
ThomasL's user avatar
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11 votes
2 answers
374 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 $[B]$ ...
Julian Rosen's user avatar
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11 votes
3 answers
395 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 ...
Haobin's user avatar
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8 votes
3 answers
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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 $ ...
D.A.G.'s user avatar
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8 votes
4 answers
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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 $ ...
Haobin's user avatar
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8 votes
2 answers
698 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+...
Mike Earnest's user avatar
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8 votes
1 answer
397 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 ...
wythagoras's user avatar
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8 votes
1 answer
499 views

Variant of Abby and Bob and 3 numbers on the blackboard

Abby selects a positive integer $A$, and Bobby selects a positive integer $B$. They then both tell their number secretly to Summy. Summy writes the numbers $5$, $8$, and $15$ on the blackboard, and ...
Hemant Agarwal's user avatar
5 votes
5 answers
471 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....
Haobin's user avatar
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5 votes
1 answer
145 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 ...
D.A.G.'s user avatar
  • 2,366
2 votes
1 answer
165 views

Replacing numbers on a board by their difference [duplicate]

The numbers from 1 to 1000 are written on a board. You are allowed to select any two numbers, erase them and write their difference on the board. If this process is repeated often enough, there is ...
ThomasL's user avatar
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0 votes
1 answer
330 views

Puzzle on a blackboard [closed]

What is the missing number in the picture above?(question is from some puzzle game) I tried some approaches, but haven't found any meaningful rule..thanks in advance!
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