Manipulating a Master TM Combination Lock.
Manipulating a Master TM combination lock consists of two main phases. The first phase consists of finding the last digit of the combination while the second phase consists of cycling through a remaining set of drastically reduced possible combinations.
Most of this info is described many places on the net.
I have added a description (Phase 2) of a technique which will greatly reduce the amount of time needed to test out a series of possible combinations along with some notes which can lead to some very minor additional possible optimizations.
A Master TM combination lock has 40 numbers (0-39) and three tumblers. This sets the upper bounds on the possible number of combination to 40 * 40 * 40 = 64,000. Trying 64K combos is not within reason but there are several manufacturing flaws in a Master TM combination lock which will help you reduce this number of possibilities to make manipulation feasable.
Phase 1 - Finding The Last Digit
Master TM combination locks have a particular flaw which allow the last number of the combo to be found relatively easily. The steps to find the last number are:
- Find the 12 stick points:
- Ignore the 7 stick points between numbers
- Note the 5 remaining stick points
- Ignore the 4 stick points which have the same last digit
The oddball stick point is the last number to the combination.
A stick point is found by attempting to pull the lock open and turning the dial. The dial might turn briefly but the lock will soon open slightly impeding dial roatation, it will stick.
After Phase 1 the number of possible combinations has been reduced to 40 * 40 = 1600. This is is much better than 64K,
but still quite a bit more than any reasonable person would waste their time trying.
Drastically Reducing the Set of Possible Combinations.
For some reason which I do not know (in order to make the assembly process simpler maybe), Master TM combination locks exhibit a relationship between the first, second and third combo numbers. This relationship is one of remainders, the remainder of a division by 4.
The first number has the same remainder as the last number
and the second number's remainder is skewed by 2 from the first and last number.
This means that by knowing the last number in the combo, we now can narrow the middle number down to 10 possibilities and we can also narrow down the first number to 10 possibilities. The number of possible combinations is now only 10 * 10 = 100!! These can be tried by anyone in under 10 minutes. By using the techniques described in Phase 2, it will probably only take about 5 minutes!
Phase 2 - Cycling Through The Remaining Possible Combos
After picking a number to try for our first combo, we need to
cycle through all the possible permutations for the possible first and middle numbers. Instead of dialing all 3 numbers for every permutation that we want to try, we can take a shortcut and only dial the last two numbers most of the time. This only works if we start our middle number permutations directly to the right of the last number and each subsequent middle number selected is to the right of the previous one.
- Try all the possible permutations sequentially by varying the first number the least frequently and only after trying every middle number possible for that first and last number combination.
- Start our middle number permutations 2 to the right of the last number
- Do not redial the first number between middle number changes until you need to change the first number.
This will result in a series of small fractional increasing clockwise and then counter clockwise movements each followed by an attempt to open the lock. A Master TM combination lock will not disturb the postion of its tumblers while attempting to open the lock.
Since dialing in a combination fresh consists of making 3 turns, entering 100 combinations would normally mean making 300 turns. With the above technique we limit 9 out of 10 attempts to 2 turn attempts. This means that it will only take (90 * 2) + (10 * 3) = 210 turns to check 100 combinations and most of these are very easy small turns.
There are still some minor tweaks to the above techniques to further eliminate the number of combinations that need to be checked. These optimizations are much more complicated to perform but for the purist they will help shave a little time off of your performance. Noting all the observations about the Master TM combination lock below, see if you can figure them out!
Observations about the Master TM combination lock construction:
- 3 tumblers, each tumbler represents one number of the combo.
- The last tumbler is connected directly to the dial. To set the last combo number, simply set the dial to the last number.
- To set the middle combo number, the dial must be turned far enough counter-clockwise to engage the middle tumbler and then further turned to the desired middle number.
- To set the first combo tumbler, the dial must first be turned far enough clockwise to engage the middle tumbler, and then further turned clock-wise to engage the first tumbler, and finally turned even further to the desired first number.
- The tumblers engage each other by a lip which extrudes from each one. The thickness of this lip affects the precision of the lock. To understand this. Turn the dial counter-clockwise more than one turn to ensure that the middle tumbler has been engaged and stop at a predetermined number such as 0. Now turn the dial clockwise until you feel/hear it engage the middle tumbler again in the opposite direction and note the number. The thickness of the lips on a Master TM combination lock should make it such that this number is now 3.
- If you extend this procedure to the first tumbler you will find that the thickness now adds up to 7. For some reason (at least on mine), the middle to first tumbler link adds 4 instead of 3.
- There are 12 stick points on the dial. A stick point is found by attempting to pull the lock open and turning the dial. The dial might turn slightly but then the lock will open slightly forcing the dial to stick.
- Of these 12, 5 will stick 'on' a number (i.e. any slack will go from the left of the number to the right of the number.)
- While 7 of these will be between numbers (i.e. any slack will not cross a number boundary, or if so, only barely).
- Of the 5 which end on number boundaries, 4 of them will share the last digit.
- The number that does not belong to the set is the last number in the combo.
- Of note on mine, but as far as I know irrelevant, if you analyze the 7 'between' numbers, of the 12 stick points, there will be three sets of 4 numbers sharing the last digit. Is this true for everyone else?
- The last combo number can be found from the stick points mentioned above.
- The middle combo number is related to the last combo number. Although there are 10 numbers each 4 apart from each other starting 2 to the left of the last combo number, i.e.:
|Last combo number||Possible Middle Combo numbers|
|0||2, 6, 10, 14, 18, 22, 26, 30, 34, 38|
|7||9, 13, 17, 21, 25, 29, 33, 37, 1, 5|
...some of these are duplicates of other combos (explained two bullets below.)
- The first combo number is also related to the last combo number. It is one of 10 (or fewer) numbers each 4 apart from each other starting with the last combo number. i.e.:
|Last combo number||Possible First Combo numbers:|
|0||0, 4, 8, 12, 16, 20, 24, 28, 32, 36|
|7||7, 11, 15, 19, 23, 27, 31, 35, 29, 3|
- The middle combo number cannot be less than 7 to the left of the first combo number. i.e.:
|First combo number||Possible Middle Combo numbers:|
|0||2, 6, 10, 14, 18, 22, 26, 30 (but not 34 or 38)|
The reason is simple, due to the thickness of the engagement lips on the tumblers.
In order to dial in a middle combo number less than 7 to the left of the last combo number, the middle dial would already have engaged the first tumbler and therefor would be changing the first combo number making it a duplicate of another combination.