CISRA Puzzle Competition 2008 - Solutions
This is the archive of the 2008 Puzzle Competition. Please visit the current competition site for information about the latest Puzzle Competition.
C.5 Cognition
The large gear with handle attached must be turned such that the red line on each small cog is pointing directly upwards towards the reference marks on the case.
From left to right, the numbers of teeth on the small cogs are:
29, 19, 11, 7, 15, 34, 23.
You may notice that none of the numbers share any prime factors. This means that the total number of unique positions of the entire mechanism (in which the teeth are aligned with the reference marks) is given by the product of the numbers:
29×19×11×7×15×34×23 = 497668710
This is a 9-digit number, and there are 9 spaces in the dials below the handle, so it seems likely that the dials are meant to display the total number of cog-teeth that the system has been turned. You may also guess this without going through this calculation.
The initial position of each small cog can be defined by the number of teeth you need to turn the main handle to make the red line point upwards individually for each cog. These positions, from left to right, are:
4, 9, 5, 2, 13, 4, 10.
Note that the left-most gear rotates in the opposite direction to all the others.
The problem can now be represented mathematically, using modulo arithmetic (or clock arithmetic), as a series of simultaneous equations:
x - 4 = 0 (mod 29) x - 9 = 0 (mod 19) x - 5 = 0 (mod 11) x - 2 = 0 (mod 7) x - 13 = 0 (mod 15) x - 4 = 0 (mod 34) x - 10 = 0 (mod 23)
The easiest approach for many solvers at this point will be to use mathematical software or a custom program to solve for x.
The smallest value of x turns out to be 265149208.
![[graphical solution]](solution-C-5_large.jpg)
Breaking up this number in the same way as the counter is broken up in the puzzle image gives:
26 5 14 9 20 8
Decoding numbers to letters using (A=1, B=2,... Z=26) reveals the answer: ZENITH.
There are also several ways to solve these equations with a calculator (or just with pen and paper if you're incredibly keen). One way is as follows:
Start by turning the handle 4 places to get the first cog aligned. We'll use t to remember how many turns we've made so far. So t = 4. In order to keep the first cog aligned while trying to also align the second cog, we can only make further turns of the handle which are multiples of 29 teeth. So, we want to find the smallest value of b, such that
b - 9 + t = 0 (mod 19) or b - 5 = 0 (mod 19), where b is a multiple of 29.
We can test progressive multiples of 29. If a number is equal to 0 (mod 19), it's the same as saying that you can divide the number by 19 to give an integer result.
1 * 29 = 29 29 - 5 = 24 24 / 19 = 1.263... 2 * 29 = 58 58 - 5 = 53 53 / 19 = 2.789... 3 * 29 = 87 87 - 5 = 82 82 / 19 = 4.315... 4 * 29 = 116 116 - 5 = 111 111 / 19 = 5.842... 5 * 29 = 145 145 - 5 = 140 140 / 19 = 7.368... 6 * 29 = 174 174 - 5 = 169 169 / 19 = 8.894... 7 * 29 = 203 203 - 5 = 198 198 / 19 = 10.421... 8 * 29 = 232 232 - 5 = 227 227 / 19 = 11.947... 9 * 29 = 261 261 - 5 = 256 256 / 19 = 13.473... 10 * 29 = 290 290 - 5 = 285 285 / 19 = 15.000
So, b = 290 satisfies our equations. This means that adding 290 turns to our turn counter, t (which was previously 4), will line up both the first and the second cogs. So we now set t to 294.
Now, we can only make turns which are multiples of (29 * 19) = 551 teeth, in order to keep both the first two cogs aligned. Using the same process as above, we can test multiples of 551 such that:
c - 5 + t = 0 (mod 11) or c + 546 = 0 (mod 11) or c + 7 = 0 (mod 11), where c is a multiple of 551.
The same method can be used to bring more and more cogs into alignment. Since the numbers of teeth on the cogs share no prime factors, we can be confident that each cog can be added to our running calculation independently of the others. Finally, we'll reach 265149208 turns, and all seven cogs will be aligned.
As above, decoding this reveals the answer: ZENITH.