42
votes
Accepted
34
votes
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
29
votes
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
24
votes
Accepted
12
votes
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
11
votes
Accepted
10
votes
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 ...
- 101
9
votes
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
7
votes
Accepted
6
votes
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
6
votes
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
5
votes
499 and the Gamma Function
$499$ and no $\Gamma$ needed, actually.
Tada!
Also without $\Gamma$:
- 16.3k
3
votes
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
3
votes
2
votes
1
vote
Explain this incorrect proof that 3=0
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 ...
- 11
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