Decoding the code

To start with 3rd digit is glued to 1

Using condition 2- 2nd, 5th, 6th & 8th can only be among 0 2 3 4. Because this is the only combination which would give a single digit number upon summation. ----- (i)

Finding 2nd 4th 6th & 10th digits

2nd cannot be 0 because then 10th also has to be 0 (cond. 6) which violates cond. 1. So 2 can be any of 2 3 4.This implies 10th can be any of 4 6 8 'respectively'. Hence the prod of 2nd & 10th can be any of 8 18 32.

Now 4th is a perfect square.1 is already occupied so 4th can be any of 4 9.

6th can be any of 0 2 3 4 (from (i)). This implies the prod of 4th & 6th can be 0 8 12-for 4th digit is 4 and 0 18 27 36-for 4th digit is 9. Overall the prod can be any of 0 8 12 18 27 36.

From cond. 7, only feasible choices for prod are 8 & 18. Out of these only 18 can be chosen because it gives four distinct values for 2nd 10th 4th & 6th digits. Prod 8 does not give distinct values for them thus violating cond. 1.

So digits corresponding to prod 18 are---2nd 10th 4th 6th <-->3 6 9 2

Now others can be determined easily.....

From cond. 5, 7th is 7.

From cond. 8, the only place left for 5 is 9th digit.

From cond. 8 and (i), only place for 4 is 5th digit.

From (i), 0 goes to 8th digit

There is just one place left unoccupied, that is 8 is the 1st digit.

So 8319427056 ANS

Now just rob the bank ! :-)

Let's start with the easiest conditions first :

Condition 4: 3rd digit is 1

_ _ 1 _ _ _ _ _ _ _ _

*Condition 3: The 4th digit is a perfect square. *

*Condition 5: The sum of the 4th and the 7th digits is a perfect square. *

This can only mean that the 4th digit is either 4 or 9; And that the 7th digit is either 5 or 7 as 4 + 5 = 9 (perfect square) and 9 + 7 = 16 (perfect square).

You now have 2 possibilities:

A: _ _ 1 9 _ _ 7 _ _ _ _

B: _ _ 1 4 _ _ 5 _ _ _ _

Condition 6: The value of the 10th digit is twice that of the 2nd digit. Condition 7: The product of the 4th and 6th digits is equal to the product of the 10th and 2nd digits.

The 10th digit can only be 4, 6, 8

• You can only use numbers 0-9.
• Multiplying any number by 2 would only yield even numbers
• Cannot be 2 because then the 2nd digit has to be 1, but condition 3 already states that the 3rd digit is 1.

The 2nd digit can only be 2, 3, 4. You get this by dividing the possible 10th digits by 2.

Next , you want to get all the possible products of 4, 6, 8 with 2, 3, 4.

Its easier to visualize if you make a table but: 10th digit * 2nd digit = product $8 \times 2 = 16$

$8 \times 3 = 24$

$8 \times 4 = 32$

$6 \times 2 = 12$

$6 \times 3 = 18$

$6 \times 4 = 24$

$4 \times 2 = 8$

$4 \times 3 = 12$

$4 \times 4 = 16$ !!! (since Cond 1: All the digits are from 0-9, and each digit is used exactly once, you can eliminate this)

So the possible products of the 4th and 6th digit has to be: 8, 12, 16, 18, 24, 32.

Next , you want to get which numbers when multiplied by either 4 or 9 (possible 4th digits, see above) will yield the above numbers (12, 16, 18, 24, 32).

$9 \times 2 = 18$

$4 \times 2 = 8$ !!!

$4 \times 3 = 12$

$4 \times 4 = 16$ !!!

$4 \times 6 = 24$

$4 \times 8 = 32$ !!!

We can get rid of the combinations (4th, 6th): (4, 2), (4, 4), (4, 8) as these combinations would make the 4th and 6th digit equal to the 10th and 2nd digits.

By the process of elimination, here are our possible candidates for (4th, 6th) & (10th, 2nd):

(9, 2) & (6, 3) both equal to 18

(4, 3) & (6, 2) both equal to 12;

(4, 6) & (8, 3) both equal to 24;

Let's put it into our list of possibilities and tackle the remaining conditions from there. Take note that we have an additional possibility C because there are two possible (4th, 6th) & (10th, 2nd) combination with 4 as the 4th digit.

A: _ 3 1 9 _ 2 7 _ _ 6

B: _ 2 1 4 _ 3 5 _ _ 6

C: _ 3 1 4 _ 6 5 _ _ 8

So are there any conditions we can use to eliminate a possibility?

Condition 8: The 5 is next to the 6, but not next to the 4. Because of condition 8 we can eliminate possibility B . 5 has to be next to 6, and B does not satisfy this.

We only have possibilities A and C remaining.

Apply condition 8 to A .

A: _ 3 1 9 _ 2 7 _ 5 6

The only remaining digits are: 0, 4, 8. The 7th digit can only be 0 or 8 as the second part of condition 8 forbids 5 to be next to 4. Hence:

A: (8/4/0) 3 1 9 (8/4/0) 2 7 (8/0) 5 6

Condition 2: The sum of the 2nd, 5th, 6th and 8th digits is a single number. Right now we know that 2nd, 6th digits are 3 and 2 respectively.

Because, $3 + 2 = 5$ , then the 5th and 8th number can only be 4 and 0. (If you use 8 as the 5th or 8th number, you will get $3 + 2 + 8 = 13$ which no longer satisfies condition 2 that the sum of 2nd, 5th, 6th and 8th is a single number .

You can't use 4 as the 8th digit because it will not satisfy condition 8 . So now we get the correct answer by substituting the values in:

A: 8 3 1 9 4 2 7 0 5 6

How do you know that A is correct and not C ? Well, we can test condition 2 to possibility C .

C: _ 3 1 4 _ 6 5 _ _ 8

Since 6 is already next to 5, we have already satisfied condition 8 . We can only satisfy condition 2 now. The remaining digits are 0, 7, 8, 9.

The 2nd, 5th, 6th and 8th digits are 3, ?, 6 ? respectively. If you add $3 + 6$ it already equals 9. Therefore, you can only add 0 to still satisfy the condition 8 single digit requirement. This would unable you to satisfy condition 8 because you are missing two digits, not one.

*With that, possibility A is the correct answer with combination: 8 3 1 9 4 2 7 0 5 6. *

simply start guessing which number can be where according to the given hints.....& you'll find 8319427056

Start: X X X X X X X X X X

Cond. 3: X X {1} X X X X X X X

Numbers whose sum shall be equal or less than 9 are marked sum.

Cond. 1: X sum {1} X sum sum X sum X X

The only possible way to have 4 digits whose sum is less than/equal to 9 is: {0, 2, 3, 4}.

Since we are already sure that {1} is in the 3rd position.

Therefore we are sure that {0, 2, 3, 4} belong to the 2nd, 5th, 6th and 8th position.

Cond. 4: X sum {1} {9} sum sum X sum X X

The only perfect squares less than 10 are {0, 1, 4 and 9}. Since we are already sure that {1} is in the third place and {0, 4} is in the 2nd, 5th, 6th and 8th place. Then {9} must the perfect square in the 4th place.

Cond 5: X {x/sum} {1} {9} sum sum X sum X {2x}

If the 2nd digit is x, then the last digit must be 2x.

Cond 6: X {3} {1} {9} sum {2} X sum X {6}

The only possible products from (2x)(x) = 0, 2, 8, 18, 32, 50. Where 2x >= 9.

Only {18} is divisible {9}. Therefore the 6th digit is {2}, x= {3}, 2x = {6} (2x*x = 18)

Cond 4: X {3} {1} {9} sum {2} {7} sum X {6}.

Only 7 can be added to 9 to make it a perfect square. (25 isn't possible since 16 is already more than 9)

The remaining unused numbers are: {0, 4, 5, 8}

1.) {5 and 8} can only be placed in non-sum positions. Since {5} is next to {6} (Cond. 7), it is the ninth position. Making {8} the first number.

{8} {3} {1} {9} sum {2} {7} sum {5} {6}

2.) Since {4} is not next to 5, then it is at the 5th position. Making {0} the 8th position.

{8} {3} {1} {9} {4} {2} {7} {0} {5} {6}

8319427056

Rob the bank! :D

Let the X be the number positions.

XXXXXXXXXX

for cond. 4

XX1XXXXXXX

for cond. 2, the possible values are 0,2,3 and 4 (sum=9)

then for cond. 3 the remaining perfect square is 9

XX19XXXXXX

for cond.7 9(6th)=(10th)(2nd)

then 0 can't be used for 2,6 and 10

and since we have 9 on the right side, that means that a factor on the right side is a multiple of 3

and with cond. 2 & cond. 6

then 9X6th = 6X3 - since you can pick 3 for the 2nd and 3X2=6 for the 10th

then 6th = 2

X319X2XXX6

for cond. 5 9+7th=perfect square for which 7th =7 since 9+7 = 16 perfect square

X319X27XX6

but cond. 8, 5 is beside 6 and 4 is not

then

X319X27X56

remaining value for 5th and 8th is 0 & 4 for cond. 2

then 1st is 8

8319X27X56

then with cond. 8 again, 4 is not beside 5 then we can rob the bank with the code below

8319427056

Shit....genius as hell....anw I really enjoyed this problem!!! Thanks!!! A fun one indeed!! (+1) @Disha Mohanty

i hav question for the thief ..... if he is able to remember this many things abt the code .. then why not the cide itself??!!! #logic

Starting with the 3rd digit...it is 1. If named a,b,c,d.... Assertions: The value that the places can take is as follows. d can take the value of 4 or 9. g can take the value of 0,5,7. d,e,g,h,i can assume 0. solution: If b=3 and d=9 then value of f=2,g=7 and j=6. The sum will be 3+e+2 +h=single digit. Assume e=4 and h=0 which will result in a single digit. The remaining values will be 8 and 5. As given 5 should be adjacent to 6.... Therefore i=5 and a=8. Result will b 8319427056

All digits from 0-9 is used once…. which means this is a 10 digit code.

After you solve all of the clues, you will eventually have one last number remaining. Since all digits CAN ONLY be used once, then just find the number from 0-9 that would be missing. Which in this case was 8.

Sum of the 2nd, 5th, 6th, and 8th digit is a single number OR <10

0 + 2 + 3 + 4 …. This is the only WAY.

2nd digit = 0 or 2 or 3 or 4 5th digit = 0 or 2 or 3 or 4 6th digit = 0 or 2 or 3 or 4 8th digit = 0 or 2 or 3 or 4

2nd digit can’t be a 0 because it doesn’t apply with the 7th rule. (The product of the 4th and 6th digits is equal to the product of the 10th and 2nd digits) Therefore, the 2nd digit would be a 2, 3 or a 4.

2nd digit = 2 or 3 or 4 5th digit = 0 or 2 or 3 or 4 6th digit = 0 or 2 or 3 or 4 8th digit = 0 or 2 or 3 or 4

From the 7th clue, we came to a conclusion that the 6th digit is a 2 for sure. Therefore…

2nd digit = 3 or 4 5th digit = 0 or 3 or 4 6th digit = 2 8th digit = 0 or 3 or 4

Combining from the 6th and 7th clue, we understand that the 2nd digit is 3 for sure. Therefore,

2nd digit = 3 5th digit = 0 or 4 6th digit = 2 8th digit = 0 or 4

According to rule number 8, 4 can’t be next to 5. Right now, for sure we know that the 9th digit is 5 because it applies with the 8th rule as well. Therefore, since 4 can’t be at the 8th digit place, it would HAVE to be on the 5 digit place. Which means….

2nd digit = 3 5th digit = 4 6th digit = 2 8th digit = 0

The 4th digit is a perfect square. 1, 4 and 9. The 3rd digit is a 1, therefore there CAN’t be a 1 for the 4th digit which leaves us with 4 and 9; both perfect square numbers.

4 OR 9

The 3rd digit is 1. Sum of 4th and 7th digit is a perfect square We know for sure that the 4th digit is 9 because it applies with rule number 6 and 7.

9 + (1 digit number that would give us a square number)

Therefore, We can either have 9 + 0 = 9 = 3 x 3 Or 9 + 7 = 16 = 4 x 4

We can’t have a 0 for the 7th digit since the “0” HAS to be on the 2nd, 5th, 6th or 8th digit place. Therefore we are left with the 2nd option.

9 + 7 = 16 = 4 x 4.

Which means the 7th digit is eventually the number 7.

The 10th digit is twice than the 2nd digit; 2 and 4, 3 and 6, 4 and 8. The 10th digit is 4, 6 or 8. The 2nd digit is 2, 3, or 4.

The product of the 4th and 6th digits is equal to the product of the 10th and 2nd digits. From the previous clue, we know that the product would either be 8 (4x2), 18 (6x3) or 32 (8x4)

Nothing goes into 8 except 4, 2, 1 and 8 itself.

For 18. 2 x 9 = 18.

Since this is the only one that applies to this rule, we can come to a conclusion that…

2nd digit = 3 10th digit = 6.

For 32 it is only divisible by 8, 4, 2, 16, 1 and 32 itself.

Therefore… 4th digit = 2 or a 9. 6th digit = 2 or a 9.

The 4th digit CAN’t be a 2 because it doesn’t apply with rule number 2. That “2” would be either be in the 2nd, 5th, 6th or 8th digit place.

Therefore, we can come to a conclusion that the 4th digit is 9 since it applies with this clue and the 3rd clue which states the 4th digit is a perfect square.

Since, 4th digit = 9. 6th digit will be 2 since it apples with rule number 2 and this rule itself.

The 5 is next to the 6, but not next to the 4.

5,6 or 6,5 5 CAN’t be next to the 4. Now using some common sense… If 6 is the 10th digit, then “obviously" the 9th digit would be a 5?

The CODE 8 3 1 9 4 2 7 0 5 6

Well it happens only in Maths, where a robber forgets the code but remembers the clue to get the number. Anyways, the code is 8319427056

Its easy 3rd no is 1 sum of 2nd 5th 6th 8th is sum of single digit so it should be 0,2,3,4 in any order 4th and 7th sum is perfect square that's why 9 and 7 respectively
10th number is twice of 2nd so 3 and 6 respectively 9th term is 5 because 5 is next of 6 now left number are 0,2,4,8 so 1st term is 8 because sum of 2,5,6,8 term is single digits Finally Answer is 8319427056

First fill the third place - 1

fill fourth place as it is a perfect square - 9, in this 4 is not possible.

fill seventh place by - 7 as sum of fourth and seventh digit is a perfect square second place's possibilities are 3, 2, 4 but product of 2nd and tenth = product of 4th and 6th. hence 2nd place can be filled as - 3

Tenth place twice of second place = 6 5 is next to 6 , therefore 9th place can be filled as = 4 fill 1st place by = 8 4 is not next to 5 therefore fill it at 6th place 8th place can be filled by the remaining digit i.e is 0

By this the code is = 8319427056

8319427056

Let the numbers be : a b c d e f g h i j 1. As told c = 1. 2. As b+e+f+h is a single no. The only possibility is sum of 0,2,3,4. 3. As d is a perfect square and c=1 and also 4 is one among (b,e,f,h) as from above condition. So d can only be 9. 4. As d+g is a perfect square and d=9 so g can only be 7. g= 7. 5. Since d f=j b and j=2b. That means 9f=2b*b. For which the only solution in the range is f=2 and b=3. So j=2b that is j=6. 6. As 5 is next to 6. So our 6 is j, so 5 will be i. I= 5. 7. As 5 is not next to 4 so h is not 4. But from point (2.) One of e or h is 4 . So e=4. And h=0. 8. So we left out with a and 8, so a=8.

Combining these our answer becomes : 8319427056

I got it...I can robbed the bank now... Step 1: 3rd digit=1 4th digit=9 -perfect square(1,4,9,16) I choose 9 7th digit=7 -(sum of 4th and 7th digit is 16-perfect square) 2nd digit=3 10th digit=6 (twice the value as 2nd digit) 6th digit=2 (product of 2nd & 10th digit is 18 and equal to the product of 4th & 6th digit) 9th digit =5(after 6) 5th digit=4 8th digit=0 (sum of 2nd,5th,6th and 8th is single number=9) 1st digit=8 (no choice-from 0-9) 8-only left

8319427056

its amazing...

8 3 1 9 4 2 7 0 5 6