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Puzzler Answer (Fahrenheit to Celcius)

Tom and Ray said that there was no algebraic solution to this puzzler. But there is. Here is the condensed version:



Variable Description Domain

F1 Fahrenheit Temp: 10s digit integer

F2 Fahrenheit Temp: 1s digit integer (1,2)

F3 Fahrenheit Temp: Decimal [0,1)



C1 Celcius Temp: 10s digit integer (1,2)

C2 Celcius Temp: 1s digit integer



Identities

F = 10F1 + F2 + F3

C = 10
C1 + C2

F1 = C2

F2 = C1



F = 1.8C + 32



solve for F1

F1 = (17F2 - F3 + 32) / 8.2



F2 = 1

F1 = 5.97560975609756 --> 6



F2 = 2

F1 = 8.04878048780488 --> 8



check

F C

61 16

82 27

0 is a digit, too

Temperatures between 4.1666…and 4.5 degrees C round to 04 degrees C and 40 degrees F

Yes but in fairness - 4C would not display on the bank sign as “04C.” Anyway - my solution still works - just change the domain of F2 and C1 to [0,1,2]

@dekabg:

And the motorcycle rider NOTICED that change in temperature, as stated in the Puzzler? Begone.

The puzzler is as follows:
Stevie’s riding his motorcycle to work when he sees a big sign displaying the temperature in Fahrenheit and in centigrade. The digits are exactly reversed. He notices the same thing on the way home. What were the temperatures?

He notices the digits being reversed. So he would have to see “O4” and then “40” - I just don’t see that a sign displaying 4 degrees C would read “04” - it would just display “4”

I don’t know whether temperature displays typically show the preceding 0 for single digit temperatures or not. I’ll have to get up early on a chilly morming or wait till next fall to find out, but I wouldn’t be surprised if they did, since it would seem be be simpler to have a display that worked the same in each place (0, 1, …, 9) rather than have a second display that showed (blank, 1, …, 9)

@dekabg:

Then why wouldn't the display show "040"?  At least on the Fahrenheit side....

Rajesh06 wrote…
Tom and Ray said that there was no algebraic solution to this puzzler.

To be fair, what they said was that they weren’t sure whether an algebraic solution existed. That’s silly, of course, because even a Course XV slacker like me knows this can be solved algebraically.

You can’t blame them for staffing out the puzzler responses. I wouldn’t want to read all of the mail from losers like us. Still, you’d think their staff would call attention to the fact that some of us provided an algebraic explanation.

My version:

Aside from cheating and just looking up the answer on a conversion table, we can solve the problem algebraically.

If we express the temperatures as combinations of digits A and B, then we can write the expression, (10A + B - 32) / 1.8 = 10B + A. (This takes into account the conversion from degF to degC, which comprises subtracting 32 and dividing by 1.8.) The expression, when solved for A, boils down to A = (17B + 32) / 8.2.

We can now plug in values for B to arrive at A, and, therefore, the temperatures that satisfy the conditions of the puzzler. However, we can constrain our choices because we know certain things – e.g., A and B can only represent single-digit integer values, the lower of the two temperature values must be above freezing because it was raining, neither digit can be zero because the display would not show a leading zero, etc.

So, we plug in ‘1’ for B and arrive at ‘6’ for A. In fact, 61 degF = 16 degC (allowing for roundoff error).

Next, we plug in ‘2’ for B and arrive at ‘8’ for A. In fact, 82 degF = 28 degC.

By the time we plug in ‘3’ for ‘B’, we are out of range for A – it is no longer a single digit.

So, the two temps are 61 and 82 degF (corresponding to 16 and 28 degC).

Isn’t that what I wrote in the initial post?

Yes, but that doesn’t make you a bad person.

The algebra is simple. Even the derivation of the algebra is simple.

Water (at sealevel, 29.92 in Hg), feezes at 32 degres F, 0 degrees C.
Water boils at 100 degrees C, 212 degrees F.

Divide one into the other (depending on which way you’re converting), adjust for the 32 degrees variance between zeros, and you have your answer.

@thesamemountainbike:

"The algebra is simple. Even the derivation of the algebra is simple.

Water (at sealevel, 29.92 in Hg), feezes at 32 degres F, 0 degrees C.
Water boils at 100 degrees C, 212 degrees F.

Divide one into the other (depending on which way you’re converting), adjust for the 32 degrees variance between zeros, and you have your answer."

I question whether your jibberjabber even qualifies as algebra, and, oh, by the way, the idea of the Puzzler was to come up with the temperatures, not the formula for conversion between Celsius and Fahrenheit (Especially since it’s common knowledge that it’s 1.8*C + 32 = F).

So, no, you don’t “have your answer.”

Why Zombie, you’ve upheld your reputation.

And you should read the question. The post was about Tom and Ray saying there was no algebraic equation. My response was that not only is there, but it’s even easy to derive.

Hey, I have an idea: why don’t you, Zombiewolf, write down the derivations of the formulas for going back and forth between C and F? Where does the 1.8 come from? Why do we add 32? How do we convert from F to C? No cheating now, no going to the internet.

Sorry @thesamemountainbike - I think you’ve missed the point. They didn’t say there was no algebra to covert from C to F but rather that there was no algebraic solution to the problem involving the reversal of digits. you addressed the former.

I see your point. Not having heard the show, I took the heading and the narrative description as posted to be the question at hand. Your “reversal of digits” combined with your and Harrons’ posts clarified the puzzler question.

I should realize by now that posts in folow up to things that were on the show often don’t give a complete picture.

Thanks for the clarification.

@the same mountainbike:

Some overzealous moderator deleted my inoffensive post wherein I corrected rajesh06 on what tomRay said. So you are wrong. You admit in a subsequent post that you were wrong. I will not hold my breath waiting for a specific acknowledgement, because that would be an unreasonable expectation given your ad hominem attack on me.

As far as the simple math goes, 0 centigrade is 32 fahrenheit. 0 fahrenheit has something to do with the freezing point of a certain mixture of water and salt, hence the 32 degree freezing point of pure water. That’s why you add 32; it’s an offset.

212 fahrenheit is the boiling point of water. Subtract the 32 degree offset from that and you get 180. Since 100 C is the boiling point of water, that’s where the 1.8 multiplication comes from.

This is common knowledge and I knew it before there even was an internet. I’ve already answered the question “How do we convert from F to C?” in the post that prompted and appears just above your snippy ad hominem post. Do you like apples? Yes? How do you like them apples?

To sum up, I am right, you are wrong.

"Why Zombie, you’ve upheld your reputation.

And you should read the question. The post was about Tom and Ray saying there was no algebraic equation. My response was that not only is there, but it’s even easy to derive.

Hey, I have an idea: why don’t you, Zombiewolf, write down the derivations of the formulas for going back and forth between C and F? Where does the 1.8 come from? Why do we add 32? How do we convert from F to C? No cheating now, no going to the internet."

Ahhhhhh, ZombieWoof. I was simply in error. You clearly have much bigger problems than that.

I wish you well up there on your pedestal.

I suppose consistency, even in the realm of the ad hominem could be considered a virtue. I suppose. Disappointed at the ease and clarity with which I met your puerile challenge, perhaps. A lesser person might be miffed, or peeved, but fortunately I am not a lesser person.

“Ahhhhhh, ZombieWoof. I was simply in error. You clearly have much bigger problems than that. I wish you well up there on your pedestal.”