## On the Subject of Quaternions

“Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.”

—William Thomson, Lord Kelvin

To disarm this module, compute the product of two quaternions as described below, and enter the correct component of the result on the keypad.

### Step 1: Computing the Quaternions

- The ten numbered keys on the keypad come in five colors: red, green, blue, yellow, and white. There are two of each color, ignoring the SUBMIT and CLR/NEG buttons.
- Along the right side of the module is the mathematical expression i
^{2}= j^{2}= k^{2}= ijk = -1. This expression indicates which color maps to which component of the quaternion. (White does not map to a component.) -
Construct two quaternions q
_{1}and q_{2}as follows:- Assign the values on eight non-white keys to the components of the two quaternions, treating 0 as 10. q
_{1}receives the number with the greater value for each component, unless the exception in Table A applies for that corresponding color, in which case q_{2}receives the larger number. - If any digit in the serial number (again, treat 0 as 10) is a component of either quaternion, multiply that quaternion’s corresponding component by -1. (Even if the same digit appears multiple times in the serial number, only negate once.)
- If there are no lit indicators, replace q
_{1}with its conjugate. - If there are no unlit indicators, replace q
_{2}with its conjugate.

- Assign the values on eight non-white keys to the components of the two quaternions, treating 0 as 10. q
- If the number of batteries on the bomb is odd, compute the product q
_{1}q_{2}. Otherwise, compute q_{2}q_{1}. This product will be used in step 2.

### Table A: Exception Rules

Color | Exception Rule |
---|---|

Red | This color belongs to the i or j component. |

Green | The bomb has at least one PS/2 port. |

Blue | The bomb’s serial number contains a letter in the word BLUE. |

Yellow | The sum of the two white keys (this time treating 0 as 0) is prime. |