Difference Cache Mystery Cache

(Cache is not at the posted coordinates. Find the right values for n and w in the equations below and the coordinates will be within a mile.)

To do this cache right, you should go to the Computer History Museum at the posted coordinates and view the difference engine that is on display through the end of 2010. Learn about how to calculate polynomial functions by using only addition (after some initial values are calculated). The machine is absolutely amazing. For anyone that appreciates a beautiful machine, I highly recommend a visit.

As an example of how to calculate polynomial functions using addition, take the equation 3x² + 5x + 3.

Calculate that f(0) = 3 and f(1) = 11. Now you figure out diff1(x) for x = 0 which is calculated where diff1(x) = (f(x+1) - f(x) for a value of 8. Now you do the same type of cascading subtractions for x = 2. So f(2) = 25 and diff1(1) = 25 - 11 = 14. And diff2(x) for x = 0 is calculated with diff2(x) = (diff1(x+1) - diff1(x) for a value of 6 (14 - 8). Now here comes the cool part. To find the solution for x = 3, you don't have to do any multiplication. You just add 6 + 14 + 25 for a value of 45 which is the solution for f(3). Figure out the diff1 of 20 and diff2 of 6 and you'll be able to compute f(4) with only addition. The equation is now f(x) = f(x-1) + diff1(x-2) + diff2(x-3). You'll note that the diff2 column always has the same value. And as you add higher order polynomial functions, you'll need more columns and more diff fuctions (like diff3).

Now that you understand how the difference engine calculates polynomial functions, you can try out different values for n and w below until you figure out the coordinates of the cache.

4n⁷ + 3n⁶ + 4n⁵ + 4n⁴ + 2n³ + 6n² + n

5w⁷ + 6w⁶ + 4w⁵ + 3w⁴ + 4w³ + w² + 5w

Bonus points for anyone that actually uses the difference algorithm above to figure out the coordinates.

You can check your answers for this puzzle on Geochecker.com.

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