Note: Cache moved and formula changed on 17 April 2016.
Poor Tom.
Ok, he is married to a supermodel, and makes $8 million US dollars per year. But he has been in a spot of bother relating to ball pressure.
See, there is this rule. The balls need to be inflated between 12.5 and 13.5 PSI (Pounds per Square Inch). Tom likes his balls on the soft side, so he has them filled to the minimum (12.5 PSI).
For years, there were never issues with this. The balls were checked at the start of the game (in the locker room), and that was that.
But then there was some suspicion that Tom's balls were softer than the rules allow. So a plan was hatched to check the balls at halftime of a key game. They were found to be below 12.5 PSI. An investigation was launched which took 4 months to complete and resulted in a 243 page report, with twenty-seven eight-by-ten color glossy photographs with circles and arrows and a paragraph on the back of each one explaining what each one was. You see, pressure drops as the temperature drops, so a ball filled at 12.5 PSI in a warm locker room will drop when used outside in colder conditions. Though the question the report addresses is: did the pressure drop by more than the laws of physics (or indeed empirical testing data) would suggest? You can find the report online and make your own judgement, but the conclusion was (in my own words):
It is possible that the deflation could have happened naturally due to the laws of physics, but more likely that someone deflated the balls after they were checked.
To help Tom prove his innocence, the Physics Department at The University of Pennsylvania (South Gloucestershire) did some analysis of data from other games. The balls were checked prior to the game in the locker room, then checked at the end of the game on the field. All temperatures are in degrees F. GT=Game temperature (on the field). LT=Locker room temperature. Elev=Elevation of the stadium, in feet above sea level. The PSI when measured in the Locker Room was 12.5 PSI in all cases. The PSI in the table is the measured PSI on the field at the end of the game. In all cases except one, the PSI measured on the field was less than 12.5.
Here is the data
| Game # |
Location |
GT |
LT |
Elev |
PSI |
| 1 |
Home |
50 |
70 |
289 |
AB.C |
| 2 |
Pittsburgh |
58 |
70 |
712 |
DE.F |
| 3 |
Philadelphia |
84 |
70 |
39 |
GH.I |
| 4 |
Denver |
24 |
70 |
5280 |
JK.L |
| 5 |
Home |
33 |
70 |
289 |
MN.O |
| 6 |
Green Bay |
-23 |
70 |
640 |
P.Q |
Assumptions
1. There was no tampering with the balls. The balls do not leak. Consider the laws of physics only.
2. The volume of the air under consideration is fixed. Note an American Football is fairly rigid, and once inflated the differences in pressure will have negligible impact on the volume.
3. The air consists of a large number of molecules, which are in random motion and obey Newton's laws of motion.
4. The volume of the molecules is negligibly small compared to the volume occupied by the air.
5. No forces act on the molecules except during elastic collisions of negligible duration.
6. When taking into account the elevation of the stadium, assume the ambient air pressure at altitude is estimated by assuming an exponential drop with altitude and a sea-level pressure of 1 atm. Disregard temperature, relative humidity, and weather conditions when considering the elevation effect. (The difference in temperature between GT and LT is however significant - but not with respect to the elevation). Note my calculations show that the official report did NOT take the elevation into account when calculating the expected change in ball pressure (Table 10), but in Foxboro this only changes the result if you go to the second decimal place. In Denver however it makes a difference. You may find this tool useful.
7. Round your PSI calculations to 1 decimal point. Do not round any interim calculations. For constants I used 4 decimal places for pressure and 2 decimal places for temperature in my calculations.
You will find physical evidence at N 51 IP.CL(G+H) W 002 BF.O(Q-M)(D+E)
Further Notes
There is no requirement to be familiar with or research the actual events which inspired this cache. All the data you require is on the cache page.
If you do want to read the full report, search the web for "Wells Report PDF". Note that it contains some strong language.
For a report questioning some of the science in the Wells Report, search for "Wells report American Enterprise Institute". This report concludes: "The evidence we present points to a simple—and innocent—explanation for the change in pressure in the Patriots footballs. The Patriots balls were measured at the start of halftime, whereas the Colts balls were measured at the end of halftime, after sufficient time had passed for the balls to warm up and return to their pregame pressure. "
Searching for "Deflategate" will give you even more to read.
The game data included on the cache page is fictional. Though the elevations are correct for the locations .
You can read the much briefer report from the University of Pennsylvania (South Glos) here: Penn Report
15 July 2016 Update: 545 days after the start of "deflategate", Tom decided to stop his legal challenge. In spite of the lack of evidence, the league commissioner has complete power to hand out discipline. Tom was suspended from the first 4 games of the 2016 season.
04 October 2016 Update: Given the small number of finds, I've added a further hint.
You can validate your puzzle solution with certitude.
Congratulations to abanazar for FTF. He has been awarded the Pennsylvania Academic Theory Scholar (PATS) degree.