Rainman

I might be necro-posting here, but I find this interesting. The problem with the "proof" is that it tries to pass off the English language as a formal mathematical language. By what standards do we decide if a sequence of words unambiguously describes a natural number? Are those standards semantic or syntactic? Let's break it down. If certain sequences of words unambiguously describe natural numbers, let S be the set of those sequences. Define the function f : S → N that maps each such sequence of words to the natural number that it describes. Your claim is that f is surjective. Your "proof" is as follows: 1. Suppose f is not surjective. Then it misses a non-empty subset of N. 2. Any non-empty subset of N has a smallest element. 3. Thus we can define n as the smallest natural number that f misses. 4. But now f(the smallest natural number that f misses) = n. So f did not miss n, a contradiction. 5. Since assuming that f is not surjective leads to a contradiction, that assumption must be false. 6. Hence f is surjective. This rewording of the OP should make it clearer. Notice how the phrase "the smallest natural number that f misses" is used semantically in step 3, but syntactically in step 4. If we did invent a formal mathematical language that allowed such syntax, then we have only used that piece of syntax once in our "proof". So we could just replace it with any other piece of unused syntax by argument of symmetry. Our language, our rules: 1. Suppose f is not surjective. Then it misses a non-empty subset of N. 2. Any non-empty subset of N has a smallest element. 3. Thus we can define n as the smallest natural number that f misses. 4. But now f(Mynd you, møøse bites Kan be pretti nasti...) = n. So f did not miss n, a contradiction. 5. Since assuming that f is not surjective leads to a contradiction, that assumption must be false. 6. Hence f is surjective. By now it is made clear that step 4 is technically just a second assumption rather than a justified conclusion.

I'd hate to disappoint them, but...

I probably shouldn't even be doing math now because I'm quite sick and feverish, but I maintain my stance that we don't have enough information to solve this problem either. We would have to know the probability distribution for French coins among all Euro coins. Imagine for example that French coins would be very rare, with only one Euro coin in a million being French. It is then much more likely that you only have one French coin and simply happened to pull it three times, than that all three coins are French which would be a one in 1018 anomaly. On the other hand, imagine that 90% of all Euro coins were French instead. Now the base probability that all three coins are French, even before you pull out any of them, is 0.93 = 0.729. Doing the test of pulling coins out and finding them French would only increase the probability that all are French. On a side note, in my last post I claimed that we could calculate P(A|B), which upon further thinking seems to be false. We can't even do that without the probability distribution.

What part of the body was measured for temperature? Different parts of the body cool at different rates.

Still, I did assume that a+b+c+d = 40. Indeed 12+11+9+8 = 40. My counterexample holds.

What are the allowed operations? Can we only use the standard operations (connect two points or extend a line segment) or may we use other tricks, for example align the straightedge along the diameter and use the other side of the straightedge to draw a parallel line?