There's a bit of confusion between owning a company and owning the shares. A company can buy shares of itself, but that does not grant it control of itself. Let's say Cute Puppies inc. has 200 shares (so 200 shares = 100% ownership). You and I have 50 shares each, and the rest is distributed among many other holders (we'll call them "the public"). So, we each own 25% of the company and the public collectively owns 50%. Now Cute Puppies inc. bought all shares held by the public, so it has 100 shares and we each have 50 shares. But a company can't control itself by definition (it still has the shares and can sell them, but it can't use those shares to vote, appoint directors etc.), so now we each own 50% of the company.