Strange Number

Logic Level 3

A strange number of 10 digits has the following properties:

The first digit contains the number of ones in the number.
The second digit contains the number of twos in the number.
The third digit contains the number of threes in the number.
The fourth digit contains the number of fours in the number.
The fifth digit contains the number of fives in the number.
The sixth digit contains the number of sixes in the number.
The seventh digit contains the number of sevens in the number.
The eighth digit contains the number of eights in the number.
The ninth digit contains the number of nines in the number.
The tenth digit contains the number of zeroes in the number.

Can you find this strange number?

The answer is 2100010006.

1 solution

Karim Fawaz
Jul 6, 2016

Let us start with all zeros in the number : 0000000000.

Since we have 9 0`s except for the 10th digit, the number should be 0000000009.

Since digit 10 is 9, the ninth digit should be 1. Since digit 9 became 1, the first digit should be 1. the number of 0s decreased by 2 and became 7, the number now becomes 1000000017.

Since digits 1 and 9 are 1, the first digit should be 2. Since digit 1 became 2, the second digit should be 1. Since digit 10 is 7, the seventh digit should be 1. The number of 0s decreased by 1 and became 6, the number now becomes 2100001006.

Since digit 10 changed from 7 to 6 then digit 6 should be 1 and digit 7 should be 0. The number now becomes 2100010006.

This should be the answer since :

We have two 1s and digit 1 is 2.

We have one 2 and digit 2 is 1.

We have one 6 and digit 6 is 1.

We have six 0`s and digit 10 is 6.

Answer is $\boxed{2100010006}$

Moderator note:

This approach offers one way to arrive at the answer. However, we are not guaranteed that this will always yield a possible solution, and we might be stuck constantly trying to adjust the values to work.

Also, how can we find all possible solutions to this problem? Hint: If $\overline{abcdefghij}$ is a possible solution, what can we say about $a+b+c+d+e+f+g+h+i+j$ ? What can we say about $10-j$ ?

Why not 1000000008?

Swapnil Das - 4 years, 11 months ago

Look at the 8. How many times does it appear anyway ?

A A - 4 years, 11 months ago

Oh no! Thanks a lot!

Swapnil Das - 4 years, 11 months ago

@Swapnil Das – This problem has one solution actually and it is enough to consider it by it's innate or a priori so to say deterministic behaveour based on the digits chosen anyway. This understanding of deterministic behaviour eventually leads to understanding the form of all the solutions possible considering the parts implied by their influence and way of determining each other form that behaviuor innately and independent or so to say by itself and therefore has to be analyzed , ting possible here anyway.

Firstly it is enough to make a general observation whose uses has the potential of arriving at many heuristic ways of proofs leading from the observation anyway. Observe the if a digit d appears for m times in the number then there will be m or m-1 digits which repeat for m times in the string for any digit d.

This understanding of how the numbers affect each other leads to the conclusion that 0 must repeat the most number of times in the string and in general , that for any such problem the smallest value should repeat the most number of times in any auto-sustainable configuration. After making this observation you can check it in the way presented in the solution, by starting from the maximum number of 0s possible and observe that because 0 itself is included in one of the position of the string it affects itself auto-referentialy.

Make the observation that any other digit which repeats more than 0 times and doesn't represent the the number of 0s in the string can't be greater than 2 because if it were greater than 2 then the number of digits necessary to construct the string will be more than 10 which is contradictory anyway.

Considering the structure of the way the nubmer affects each other so to say the string will necessarily have the digits 0 , the digit which represent the number of times 0 repeats , the digit which represents the digit before in the string and so on , in a way which will eventually close auto-referentialy anyway so to say. The number of times 0 repeats is greater than 2 and there is only one number greater than 2 which repeats 1 time with other repeating for 0 times meaning that digit 1 shall repeat 2 times which completes everythign and makes it auto-sustainable. Because the number of digits different from 0 is bounded by the expression 10 - n , n being the number of 0s and the only possible solution whcih is sustainable as seen a priori uses 4 digits it means the number of times 0 repeats is only 6 with the other digits being 1 1 2 6.

This is a very rushed proof. I tried to make one some time ago but couldn't express even in this very rushed way what I wanted to say anyway.

A A - 4 years, 11 months ago

Your solutions describes a procedure by which a solution to the problems is found. Nonetheless it doesn't justify why this procedure of starting with 0s and replacing them to adjust the string works and as such doesn't also guarantee there is no other solution. Therefore , can you justify how to arrive and why the procedure works ? In this way you not just show a solution that works but explain the principles from which it is derived.

What is the inner structure that makes your procedure work should be explained. I think that will complete your solution and understanding of it and cute problem anyway.

A A - 4 years, 11 months ago

Hi,

What I am using here is by trial and error. This may not be the best way to solve it or there may be a better solution. This is the way I used it to solve this logic problem and it produced the solution and when I checked it at the end, I found that it was correct.

The reason why I started with all 0's as the original guess because, in the beginning, I don't know what is the repetition of every number so I used these zeros as the original trial and it worked with me.

As far as I know, this is the unique solution which exists. I did not try other ways or to find other solutions, if you have another method, you are welcome to post it.

Thank you for your comment.

Karim Fawaz - 4 years, 11 months ago

ah , I understand what you mean. I think I may have some other solution for the problem but I will have to chissel it and also see if it is indeed correct.

My way of solving it was also some sort of trial and error with some reasoning. That is the procedure isn't systematic in the first approach of mine anyway.

Yet I guess there is some way of solving it and think I saw something which might lead to it anyway , by considering it's structure.

Nonetheless , the fact that that procedure of starting with 0s worked still anyway implies that there might be something that makes it work. It should be therefore made explicit what is that thing if there is indeed some reason behind it. Note nonetheless that it can be just a coincidence though I doubt it. And thanks for your reply and comment too anyway.

A A - 4 years, 11 months ago

Strange Number

The answer is 2100010006.

1 solution

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