# Tag Info

Accepted

• 446

### Explain this incorrect proof that 3=0

The problem is substituting $x=1$ into the original equation. In the first part, we show that: If $x$ is a root of $x^2+x+1$ then it is also a root of $x^3-1$. Then in the second part we assert: ...
• 17.2k
Accepted

### Do Langford squares exist?

Langford squares are not possible. Consider the middle two rows of a $2n \times 2n$ Langford square. They, like all rows, must contain $n$'s. Any $n$ must have a partner $n$ in its column that's $n+1$ ...
• 24.5k
Accepted

• 115k

### Explain this incorrect proof that 3=0

Substituting one equivalent equation into another generally produces extra roots. As a degenerate case, substituting an equation into itself gives a tautology, but that doesn't mean the original ...
• 2,456
Accepted

### Creating a clever hemisphere

This is a standard application of the pigeonhole principle.

### Explain this incorrect proof that 3=0

All the steps made except of the substitutions are in fact valid equivalences, meaning that the equations before and after doing the transformation step have the same solutions. For example, the ...
Accepted

### Do non-trivial Skolem squares exist?

No non-trivial Skolem squares exist. First, observe that any number $k$ appearing in a Skolem square must be part of an axis-aligned square of width and height $k$ whose four vertices are all $k$. For ...
• 24.5k
Accepted

### 2x8 Langford Rectangle

Here is one of the possible answer: the first thing is to put The rest
• 29.6k

### Explain this incorrect proof that 3=0

IMHO1: this is not really a problem about algebra, it is a problem about logic. IMHO2: while other answers have explained something about the logic, I can't see that it has been clearly stated what OP ...
• 399
Accepted

### The "Slightly Spooky Sequence" Game

Shortest and longest games? Who wins this game with a limit of 30? Length of longest/shortest games and who wins with optimal play for N from 1 to 200
• 11.3k

### 499 and the Gamma Function

$499$ and no $\Gamma$ needed, actually. Tada! Also without $\Gamma$:
• 16.3k

### Save now! All the digits at half the price

A bit messier than I'd like, but here goes. As other answers have noted, the doubleable PD10 numbers we want are those where Let's show that every such number can be specified by: for an overall ...
• 24.5k

### Explain this incorrect proof that 3=0

• 2,531
The original $LHS = (x+ 0.5)^2 + \frac 34$, which is always greater than $0$, so the equation has no solution. Dividing by x doesn't affect the equivalence of the new equation, so the next equation ...