*Inspector Maths uses mathematics to fight crime. *

*It’s not very effective.*

Inspector Maths stood silently in the deceased’s small house, surveying the sorry scene sibilantly.

There were beer and bits of brains splattered all over the kitchen floor, courtesy of the

unsuccessful merger of a case of beer and the deceased’s skull.

Clearly, Billy the dwarf had been in the process of removing the beer from the fridge-freezer, when he had slipped (most likely on the rollerskate, perhaps ironically the source of his erstwhile livelihood, which now lurked guiltily in the opposite corner of the room) and banged his head quite badly just before the entire case of beer went through it.

Inspector Maths knelt down and dipped a fancy-as-fuck thermometer in a small puddle of beer, checked her watch, and then made a note of the temperature setting on the fridge. As she did so, she mused upon the sick contrivance of a dwarf having a full-size standing fridge-freezer unit. It was just that kind of neighbourhood.

The idea was that the pool of beer would be gradually warmed by the temperature of the room, and knowing the temperature of the room and the original temperature of the beer would enable Inspector Maths to estimate the time at which it had been spilled, ie. the time of Billy’s untimely demise, using a simple first order differential equation.

The coroner would have performed similar calculations earlier, by sticking his own

thermometer up the deceased’s cooling bum, but he was becoming terribly fond of sticking things up people’s bums and Inspector Maths didn’t want to encourage him needlessly by asking for his results.

Letting **T(s)** be the temperature of the beer **s** hours after Billy’s fall, and letting **R** be the room temperature, which, thanks to Billy’s surprisingly sophisticated aircon unit, was constant, Inspector Maths planned to investigate the rate of change of the temperature of the beer, and thus, estimate the time at which the unfortunate accident had occurred.

The room temperature was set at 24 degrees, the fridge temperature set at 4 degrees, the beer warmed to a tepid 19 degrees, and the time, at least for now, was 14:27.

At first, the beer would have warmed rapidly after being taken out of the fridge, with the heat change slowing as it approached 24 degrees, and finally stopping when it reached room temperature.

This rate of change in temperature at time **s** was described by

**dT(s)/ds = k(R-T(s))**

where the **k** mentioned was some constant accounting for certain properties of the beer and its interaction with the air; e.g. how much of the beer’s surface area was exposed to the air, its heat-conductivity level, and other trifling matters with which Inspector Maths did not concern herself, because she wasn’t a fucking physicist.

From this, Inspector Maths was able to derive an equation (no pun intended) describing the temperature as a function of time.

After she had Snapchatted her calculations to an undergrad she was trying to impress/sleep with (neither of which explicitly implied the other), she realised that she had introduced equation (III) completely unnecessarily, but it was too late to do anything about it now. He wouldn’t notice anyway, he was studying Arts of some description. It probably made her look smarter. Maybe superfluous equations were the way to go.

Anyway, she had her formula:

**T(s)=R-(R-T(0))e**^{-ks}

Filling in the available information, that gave

**T(s)=24-(24-4)e**^{-ks}

There was just that pesky **k** to deal with, and that was a constant, so all Inspector Maths needed was a little time. And since she was on the clock, that was fine by her. She lit a cigarette and waited.

A thought occurred to her. She smiled and scribbled in her notepad.

“*Tell Peterson his momma is so fat that one can not reasonably assume that the room in which she dies will act as a thermal reservoir.*”

After an hour had passed, she took out her fancy thermometer, which had never been up anyone’s bum, and she measured the temperature again. 20 degrees.

These measurements allowed Inspector Maths to perform some brief calculations to see how long it had been since the death/spillage/what have you.

It seemed Billy had died at about 08:14.

Inspector Maths cursed. She bent down and took an intact can from the case of beer on the little man’s head. It was going to be a long day.

Normally, this accidental, albeit somewhat comical death, wouldn’t have been worth getting out of bed for. She and the coroner shouldn’t even be here, stealing beer and sticking thermometers up bums. She certainly wouldn’t have cared what time the dwarf had died at.

But Inspector Maths was investigating a murder. And it seemed her prime suspect, Billy the amazing rollerskating dwarf, had a very solid alibi.

DUN

DUN

DUN!!!!