How to unlock 4-digits lock combination ?

Number Theory Level 4

Your suitcase has FOUR digits lock combination $A|B|C|D$ . Due to a mechanical problem, the gears of the lock get stuck together. and cause the following problems:

When you move gear $A$ one step, it causes each gear attached to digits $A|B|D$ to increase by $1$ .

When you move gear $B$ one step, it causes each gear attached to digits $B|C|D$ to increase by $1$ .

When you move gear $C$ step, it causes each gear attached to digits $A|C|D$ to increase by $1$ .

When you move gear $D$ , it has no effect at all.

If the FOUR digits right now are on $\large 1087$ and to open the suitcase you need to set the lock to $\large 7117$ by using only gears $A, B,$ and $C$ . If you are going to move $A$ by $a$ steps, and $B$ by $b$ steps and $C$ by $c$ steps, enter your answer as $\ a +b+ c$

The answer is 20.

2 solutions

Anirudh Sreekumar
Jan 18, 2017

1)Since turning any of the gears will increment the position of D ,the total number of turns have to be a multiple of 10.(as the digit on d is already correct)

2) Move gear C, 2 steps, now the digits on B and C match up (are 0 ) and A is at 3.

3)Now make the digit on A equal to 7 by alternating turning the gears A and C twice each, so that the digits on B and C still match up.(are at

4)Rotating each of the gears A and C, 5 more times,the digit on A remains the same after cycle of 10 and digits on B and C are at 7.

4)Turn gear B 4 times it doesn't affect digit of A and all the digits are at the required number 7117

and it takes us 7 turns of gear A,9 turns of gear C and 4 turns of B so the answer is 7+4+9=20

Ossama Ismail
Jan 4, 2017

Answer Answer $= 7+4+9 = 20$

$a = 7 , \ b = 4 ,and \ c = 9$

Solve the following equations to get the answer:

$(a+c+1) \%10 = 7$
$(a+b+0) \%10 = 1$
$(b+c+8)\%10 = 1$
$(a+b+c+7) \%10 = 7$

Solve the following equations to get the answer:

How did you do it?

Pi Han Goh - 4 years, 5 months ago

This problem needs a complete search technique to the find the answer, but here a simple trial and error will solve it : from the given equations possible answers can be written as follow:

a + c = 6 , 16 or 26

a + b = 1, 11, or 21

b + c = 0, 7, or 27

a + b + c = 0, 10, or 20

from the possible solution you will get 7,4, and 9...

Ossama Ismail - 4 years, 5 months ago

@Ossama Ismail How did u get the following equations?

Kaustubh Miglani - 4 years, 5 months ago

These equations are obtained from the intial and the final status of each gear.

e.g. If the initial digit is 3 and the final state is 8, this requires the gear to be advanced for 5 steps or 15 or 25. ... etc. ( this is rotary gear so if you advance it by $n$ steps it will reach to (n + initial state )%10.

Ossama Ismail - 4 years, 5 months ago

How to unlock 4-digits lock combination ?

The answer is 20.

2 solutions

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