Digit adding problem - 2

Number Theory Level 2

We can add up multiple digits in a multi-digit number to get a single-digit number.

For example, 2354 has 4 digits- 2 , 3 , 5 , 4 ; We have to add the digits in this number TWO times to get a single digit number; 1st step. 2354 : 2 + 3 + 5 + 4 = 14 ; 2nd step. 14 : 1 + 4 = 5 .

True or false: there is at least one 4 -digit number we have to add FOUR times in the above way to get a single-digit number.

False True

Sagarchandra Roy by Sagarchandra Roy , 26, Bangladesh

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2 solutions

Giorgos K.
Feb 18, 2018

There is no such 4-digit number, otherwise this Mathematica code would print it!

For[s = 1000, s < 9999, s++, t = 0; j = s; While[Length@IntegerDigits@j > 1, j = Tr@IntegerDigits@j; t++]; If[t > 3, Print@s]]

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Interesting. I believe you. Whenever i do not understand the whole code. Nor i cannot know if it is Python or another program language. But i know that if you publish it, you have checked it, and corrected every program error ; and that among members, many can program in the same language as your tool, and they detected no error. But philosophically (?) how can any of us be sure that no mysterious number passed your mechanized filter without the machine detect it - the problem is larger, for example i wrote a "mathematical" proof, anyone can say "this is not precise enough", but i believe myself, and many believe me, because i use words and logic that are "proof" in our minds - already used and tested and tried logic - but we cannot count how many ideas can escape to us believing in logic. Language and logic are ancestors of machines. How being absolutely sure ?

Leonblum Iznotded - 2 years, 10 months ago
Leonblum Iznotded
Jul 23, 2018

The number(s) having the biggest number of steps to get the "trace" (the one-digit result) are the number(s) having the biggest carry.

So it is natural to try with 9999, the biggest 4-digit number, among the biggest carry-maker :

9999 -> 9+9+9+9=36 -> 3+6=9. Only 2 steps.

No number 4-digitted (in decimal system) need 4 steps to get a one-digit trace. Every 4-digit is traced within 1 or 2 steps.

1 Helpful 0 Interesting 0 Brilliant 1 Confused

Your conclusion is not true, I'm afraid - for example, 1099 → 19 → 10 → 1 1099 \rightarrow 19 \rightarrow 10 \rightarrow 1 needs 3 3 steps. In fact there are no 4 4 -digit numbers that need 4 4 steps, but just looking at 9999 9999 doesn't prove it.

Chris Lewis - 1 year, 8 months ago

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