On the Subject of SI-HTS

The better you are at physics, the worse you’ll be at SI-HTS.

The year is 3580. You are the president of the United States, and are currently in a heated political debate with the president of the East Asia Coalition. As with all great debates, the conversation is being settled via a best of seven coin toss tournament.

This last coin toss will decide the fate of the planet. Letting either side win would lead to a nuclear disaster and the downfall of humanity; You must therefore force a draw to keep civilization afloat. How do you force a draw in a best of seven, you ask? You must simply land the coin on its side, counting as a point for neither party.

The Game So Far

First, determine the previous coin results by interpreting each character in the serial number as base-36, and take that number modulo 5. Interpret a 0 or 1 as Heads flip, a 2 or 3 as a Tails flip, and a 4 as a Side flip. Then, if the difference between the number of heads and tails flipped is greater than one, cycle the numbers until it is not (1,2 = H, 3,4 = T, 0 = S, then 2,3 = H, 4,0 = T, 1 = S, etc.). If none are valid, the score is 3 heads, 3 tails. The United States is represented by the Heads result. Based on the current state of the game, you must influence the next coin toss in such a way that will tie the game.

The Coin Toss

Next, you must find how this toss will play out without influence. The opponent is up, so you cannot control the initial trajectory. Luckily you have come prepared with information about your opponent, on top of studying each and every throw leading up to this one. On the module, the Height and Power of your opponent are listed, which you can use to determine the trajectory of the coin. Unfortunately for you, your opponent also knows their stats, and will toss the coin in a way that is more likely to land in their favor. They will either flick the coin, or underhand toss the coin.

	Initial Velocity	Initial Height	Coin RPS
Underhand	(1/2) * PWR m/s	(2/5) * HGT cm	(1/3) * PWR RPS
Flick	(2/5) * PWR m/s	(1/2) * HGT cm	(5/3) * PWR RPS

Rigging the Game

In the previous rounds, you have interfered with your opponent’s toss with a simple trick involving magnets that can either increase or decrease the acceleration of the coin*. Your opponent has caught on to your sneaky tricks, however, and will choose the toss with the highest expected value. To calculate your opponent’s expected value of each toss type, take the average of the result with no interference, with increased acceleration, and with decreased acceleration. You can note a tails outcome as +1, a heads outcome as -1, and a side outcome as 0 in this context.
*In reality, the magnitude of the magnetic force is based on the velocity of the coin. We will ignore this.

	No Interference	Increased Acceleration	Decreased Acceleration
Acceleration	-9.81 m/s2	-9.91 m/s2	-9.71 m/s2

In order to solve for the outcome of a given toss, first put together the equation for the coin’s position as a function of time as follows:

y(t) = y₀ + v₀t + ½at²

where ‘y0’ is the initial height, ‘v0’ is the initial velocity, and ‘a’ is the acceleration due to gravity and optional magnetic interference.

Then, solve for t when y(t) = 0 using the quadratic formula; this is when the coin will hit the ground. There will be two answers; take the positive one. Finally, multiply the RPS by the time received in the previous step to identify how many rotations will have occurred when the coin lands. Take this number modulo 1, and use the following chart to determine the result:

Number	[0.0,0.22)	[0.22-0.28)	[0.28-0.72)	[0.72-0.78)	[0.78-1.0)
Result	Heads	Side	Tails	Side	Heads

A Tails is a +1 for your opponent, a Heads -1, and a Side 0. Take the average of all three results for one toss type. This is the EV for that toss type. Your opponent will choose the highest EV toss type, defaulting to flick in case of a tie.

NOTE: If the silhouette is inverted, your opponent is the Australian Prime Minister. If you change the acceleration on the coin, make it the opposite to account for the Coreolis Effect.

Appendix: Quadratic

Quadratic formula:

For a polynomial of the form ax² + bx + c = 0,
the solutions can be derived from the expression:
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$

Make sure to check your units! 100 cm = 1 m.