# Tag Info

Accepted

### Averaging numbers on the blackboard

First choose $2014$ and $2016$. Average = $2015$. Now take the $2015$s. Their average is $2015$. Now choose $2015$ and $2013$. Average = $2014$. Choose $2014$ and $2012$. Average = $2013$. Note ...
• 1,348
Accepted

### 99 numbers on the blackboard

I believe this is the answer. The strategy is below. Edit: Slightly clearer response with strategy.
• 1,016
Accepted

### Make a certain number with an unusual calculator

Let $A(x)=2-\frac{1}{x}$ and $B(x)=1+\frac{1}{x-1}$. Notice: In total, the sequence of button presses is (One continued fraction representation of $\frac{1003}{100}-1$ is $[9;33,2,1]$. This is not a ...
• 33.7k

### Averaging numbers on the blackboard

Reasoning: After this we... ...going all the way back to...
• 709
Accepted

### Blackboard problem with 2016

Obviously we're never going to get an irrational number. So any number we get can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers ($q$ can be $1$, if the number is an ...
• 33.7k
Accepted

### Blackboard problem with polynomial

The smallest possible value of $n$ is Claim: We can get every non-negative integer $n\leq 2016$ on the board. Proof: By induction. We start with $n=0$ on the board. We can get $1$ using Lord of the ...
• 14.3k
Accepted

• 18.6k
Accepted

### Aatif averages numbers on the blackboard

The answer is : Explanation : Generalization :
• 7,165
Accepted

### Variant of Abby and Bob and 3 numbers on the blackboard

The first ring happens before any information is passed, so we can't draw any conclusions If the second ring happens, If the third ring happens, If the fourth ring happens,
• 19.7k
Accepted

• 27.5k

### Blackboard problem with 2016

Obviously we have $x=2016$. Let $x=2$, then we can write $\dfrac24$. With $x=4$ we can write $\dfrac46$, etc... So any even number $\lt2016$.
• 35.6k

### Make a certain number with an unusual calculator

I wanted to post my solution as well, because it looks a little diffferent than f'''s. If we press $[A]$ repeatedly we see $\frac{3}{2}$, $\frac{4}{3}$, $\frac{5}{4}$, $\ldots$, and indeed $[A]$ ...

### Blackboard problem with 2016

I tried tackling the problem numerically. I found that up to 13 layers of depth(the furthest I could go), there was no solution. First I inverted $f(x) = 2x+1 -> f^{-1}(x) = \frac{x-1}{2}$ \$ ...
• 1,867