## Puzzle #153: Get Your Gifts

Another nice puzzle from the NASA collection sent to me by Suman Saraf.

Four friends, Holly, Belle, Carol, and Nick, gather for May birthdays. Holly announces that she has a game before dinner. She hid gifts for each of her friends inside three separate boxes secured with padlocks. She challenges her friends to figure out the combination without consulting each other.

She provides the following information. All the padlocks have the same combination. The padlocks use 3 digits from 0 to 9. She also tells them that the sum of the three digits is equal to nine, and every digit is equal to or greater than the previous digit. Holly tells each of her friends one of the digits in the combination. She states, “I’ve given the first digit to Belle, the second digit to Carol, and the third digit to Nick.” The caveat is that the friends cannot share their numbers with each other or they will forfeit the gifts.

Then Holly gives her friends 30 minutes to open the padlocks while she watches and finishes dinner.

The three friends begin to think of the solution. One by one, they each try their hand at their padlock, but none of them opens the padlock. Seeing that no one has succeeded, suddenly Carol realizes she knows the answer, and successfully opens her box, revealing a new fitness tracker. Following this, Nick opens his padlock, revealing a new tablet; and Belle opens her box to find new pair of headphones.

Having watched this entire event unfold, can you determine the correct combination?

As always, please send your answers as comments within the blog (preferred), or send an e-mail to alokgoyal_2001@yahoo.com. Please do share the puzzle with others if you like, and please also send puzzles that you have come across that you think I can share in this blog.

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### 14 Responses to Puzzle #153: Get Your Gifts

1. If Belle had 3, then Belle would have known that the number is 333 and the problem would have been solved. So after first second, both Carol and Nick know that Belle has less than 3. If Carol had 1, then she would not know whether Nick has 7 or 8. But Since Nick also did not open the padlock – Carol knows that the sequence is 018.

• Praneeth says:

if the sequence is 018 , Nick must be the one to open the locker first as there is no other alternative to 018 when the third number is 8.

• Vishal Poddar says:

The possible outcomes are 009, 018, 027, 036, 045, 117, 126, 135, 144, 225, 234, 333. Belle was not able to answer, so the number is not 333. Carol was not able to guess so the number is not 009. Nick was not able to guess, so the number is not 018. After Nick was not able to answer the number Carol was able to find the number, this implies that there were 2 possibilities from the second number. As Carol was able to find the number, this means that the second digit is 1 and if it is not 018, then the number is 117.

• Oops. Sorry. My screw up. It has to be 117.

• Alok Goyal says:

Pratik, after ages you may have something wrong here!

2. Vishal Poddar says:

The possible cases are 009, 018, 027, 036, 045, 117, 126, 135, 144, 225, 234, 333. Belle was not able to guess, it rules out 333. Then Carol was not able to answer, so it rules out 009. Then Nick was not able to guess, so it rules out 018. Carol answered after Nick was not able to answer, this implies that there were 2 possibilities from the second number. Carol knew the answer after ruling out 018, it means that the second number was 1. So the possibility left with Carol is only 117.

3. Vishal Poddar says:

The possible outcomes are 009, 018, 027, 036, 045, 117, 126, 135, 144, 225, 234, 333. Belle was not able to answer, so the number is not 333. Carol was not able to guess so the number is not 009. Nick was not able to guess, so the number is not 018. After Nick was not able to answer the number Carol was able to find the number, this implies that there were 2 possibilities from the second number. As Carol was able to find the number, this means that the second digit is 1 and if it is not 018, then the number is 117.

4. Karan says:

Combinations that can be eliminated
333(Belle),009(Carol),018(Nick)
Now since Carol got the solution, the combination’s second digit must be exclusive, which is possible with the number 1. Since 018 had been eliminated the remaining combination is 117. 117 is the combination.

• Praneeth says:

if it is 117, Nick does have only two combinations 117 and 027 when the third digit is 7 so once nick knew that none of them were able to open in first chance, she will be left with one combination from the two and she has to open at the same time as carol i.e nick doesnt need the information of carol opening the door but in the question nick uses the information of carol opening the door.

So I think it cannot be 117.

• Alok Goyal says:

Very nice!

5. Praneeth says:

I think the answer is 045 because
Nick should have atleast three combinations with the third digit he has because Nick uses two sets of informations 1) nobody is able to open the door in the first attempt and 2)Carol is able to open in the second attempt. So Nicks third digit needs to be 5 as any other number in the third digit gives only two combinations or less. So Nicks options are 045,135 and 225.

Belle should also have atleast three combinations which leads to 0 or 1.

Now Carol should have a combination with exactly two combinations using the second digit i.e possible with only 4 on the second digit leading to give us options as 144 and 045.

Now a possible combination which can satisfy all three people is only 045.

6. Praneeth Kumar says:

Sorry, 045 is incorrect , I agree with 117.

7. Abhinav Jain says:

117

8. ASHISH JATAV says:

According to question everyone tried their hand to open the lock at first. it is not mentioned in the question thet how many combinations they can use to unlock in first attempt. so my argument is that, everyone can unlock in their first attempt only as there are only 10 possible combinations
0 0 9
0 1 8
0 2 7
0 3 6
0 4 5
1 1 7
1 2 6
1 3 5
2 2 5
2 3 4
3 3 3
additionally every one has one digit also so they will require less combinations to check.