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## Numb3rs 110: Dirty Bomb
## Triangulation
In this episode a truckload of radioactive waste has been hijacked and Charlie uses triangulation of the radiation the waste emits to find where it is. This is mathematically similar to trying to find a lightbulb in a very large field (without moving). If you are standing in a field, then you will be able to see the lightbulb but you won't be able to tell how far away it is. This means you know that it lies somewhere on a particular line that goes through you, which probably wouldn't be particularly useful, since to find the lightbulb without gathering more information you would have to walk along the entire line to get to the lightbulb.
However, let's say you have a friend in the field and both you and your friend have lightbulbs and radios. Then you can report to each other the angle between the friend's lightbulb and the other lightbulb. Is this enough to find the other lightbulb? Not quite, because there is no angle-angle theorem in geometry. In other words, if you adapt coordinates so that you are at the origin, then if you double the distances of the other lightbulb and your friend from the origin, both the angles that you and your friend measure will be the same. The only way to fix this is to measure the distance between you and your friend.
Of course, if you are unlucky there will be a problem with this. There's a chance that the lightbulb will lie on the line between you and your friend. If this is the case, it would be impossible for you and your friend to tell where on the line the lightbulb is. This can be fixed, however, by adding a third friend. If the three friends make sure they aren't all standing on a single line, then they can always find the lightbulb.
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Prisoner 2 Testifies | Prisoner 2 is Silent | |

Prisoner 1 Testifies | 8, 8 | 0, 12 |

Prisoner 1 is Silent | 12, 0 | 2 months, 2 months |

- If we have the following table, what are some conditions on the numbers
*a, b, c, d*so that the argument given above still works and the equilbrium solution is the lower right hand square? Prisoner 2 Testifies Prisoner 2 is Silent Prisoner 1 Testifies a, a b, c Prisoner 1 is Silent c, b d, d - Use reasoning similar to the argument above to figure out the equilibrium choices for the following game. (The left number is the payoff for player 1, and for this game bigger numbers are better.) Player 2 choice A Player 2 choice B Player 2 choice C Player 1 choice A 4, 9 6, 4 1, 9 Player 1 choice B 5, 3 9, 5 5, 2 Player 1 choice C 1, 7 15, 12 10,8
- Will similar reasoning work for the following game? Why or why not? Is there an equilibrium solution? (An equilibrium solution is a choice for player A and B such that given knowledge of player B's choice, player A wouldn't want to change his choice, and vice versa.) Prisoner 2 Testifies Prisoner 2 is Silent Prisoner 1 Testifies 2, -2-2, 2 Prisoner 1 is Silent-2, 2 2, -2

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