Can you solve this problem without algebra?

“The sum of two numbers is 464 and the difference is 212.” – aren’t both of those statements algebra? Maybe the answer is no for me, because I know those as two algebraic statements? They taught us maybe 8th grade that sentences like that are equivalent to algebraic formulas.

I’ll try completely ignoring and forgetting those two and reading your story problem version:

My first reaction is how do I know more sophisticated facts (i.e. If you win the bet you will end up with $464 and if you lose you will end up with $212.) than the trivial fact of how much money I have?

Also yeah I can answer the first question in a real situation like this, without algebra by counting my money. I could probably answer the second question by asking the dealer what the bet is.

I also think that your problem’s implied notion that any “bet” must be a matter of +/- a singular bet amount, is both logically false, and untrue of actual bets. It’s quite normal for a bet to have various amounts you can win or lose depending on what happens. For example, it could be that the bet is something where I will only lose $10 if I lose, but I can win $242 if I win. That’s not even an uncommon type of bet. In that case I have $222.

So I assert your story problem version is unsolvable as written – you’d need to explain what the actual bet is like.

If you explained that it’s a bet where you either lose an amount, or win the same amount, and it happens in one bet, yeah it’s obvious to me that the amount I have is half-way between the two numbers, so the distance between the two numbers is found by subtracting the smaller number from the larger one, and the bet amount would be half that.

The number line idea is something I get immediately.

I would just take a guess at two numbers that are 212 apart like 110 and 312, and ok that would be 422, so I am too low.

Since it is low I would go higher maybe 122 and 334 which equals 456, getting close.

I see now I need the last number to be 6 and 8 to get a 4 at the end, so I would try 126 and 338 equals 464.