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Scott Cram Inner circle 2678 Posts |
On another board, I came across a discussion about the fact that -40 degrees is the same in Celsius and Fahrenheit temperature scales, and how interesting it was that the common temperature was an exact integer. From there, I developed the following puzzle. I don't have an answer for it yet, so hopefully this will provoke some thought.
Puzzle: A new country, named Madeup, forms. Part of the way they decided to show their independence and revolutionary spirit is by developing their own new temperature scale. It's not in degrees Celsius, and it's not in degrees Fahrenheit. Instead, it's in degrees Madeup (or degrees M, for short). Now, even though the new temperature system is different from the other two major systems, they're still human beings, so their new temperature system does have some similar features as the existing systems: • The new system is based on one temperature that represents the freezing point of water at sea level, and a higher number (not lower, not the same) for the boiling point of water at sea level. • Both the freezing point and the boiling point are set as exact integers. • Both the freezing point and boiling point are somewhere within the range of 0 degrees M to 300 degrees M. (Obviously, if 300 degrees M is the boiling point of water, the freezing point of water must be 299 degrees M or less. Similarly, if the freezing point of water is 0 degrees M, the boiling point must be 1 degree M or more.) Questions: Under these circumstances... 1) ...what is the probability that degrees M and degrees C have a temperature that's represented by the same number in both systems AND that's it's an integer? 2) ...what is the probability that degrees M and degrees F have a temperature that's represented by the same number in both systems AND that's it's an integer? 3) ...when the nearby island of Unreal develops their own temperature scale in degrees U (for Unreal, of course), with the same characteristics described above, what is the probability that degrees M and degrees U have a temperature that's represented by the same number in both systems AND that's it's an integer? If someone can find a definitive answer to these questions, discovering the parameters that result in a common integer temperature could interesting. |
Scott Cram Inner circle 2678 Posts |
Just to help move this puzzle toward a solution, it's probably easiest to think of the two temperature systems as two lines on graph paper.
Think of one of the temperature systems as a "reference system", and use its freezing temperature as our first x coordinate, and its boiling temperature as our second x coordinate. Our first y coordinate would be the freezing temperature in whatever system we're comparing, and our second y coordinate would be the boiling temperature in whatever system we're comparing. For example, let's choose Celsius as our reference system. Freezing is 0, and boiling is 100, so when we plot lines, we'll make sure the line goes through (0, y1) and (100, y2). The variables y1 (freezing) and y2 (boiling) for both coordinates will be provided when we compare Celsius to another system (or itself). If we compare Celsius to itself, then our line must go through the points (0, 0) and (100, 100). The line equation there is pretty simple: y=x (That makes sense - Celsius is equal to itself.) If we compare Celsius to Fahrenheit, then our line must go through the points (0, 32) and (100, 212). This is a little trickier to work out, but if you understand the math, the formula for the line works out to: y=(9/5)x + 32 (I wonder if anyone else has hit on that formula? ) So, if we want to find where the common temperature is, we're effectively looking for where the lines meet. We want to know where y=x and y=(9/5)x + 32 give the same number - in other words, where x=(9/5)x + 32. I've set up this equation in Wolfram|Alpha so that it works with any temperatures. Just set f to the freezing temperature of system 1, g to the boiling temperature of system 1, m to the freezing temperature of system 2, and n to the boiling temperature of system 2. The rest of the variables will be calculated for you. (s is the slope of the line, and x is the common temperature between the two systems.) I've also set this up in the Desmos onl......lculator, so as to provide an interactive and visual version of the problem, as well. (Click on the help button in the upper right corner, if needed.) It's not hard to discover that when both systems have the same "spread", that the lines don't cross. If degrees M has the freezing point of water at 27 degrees M, and the boiling point at 127 degrees M, that's a 100-degree spread, the same as Celsius, so they have no temperature in common. Any other discoveries that could move this puzzle along? |
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