Interesting Math Puzzle - In Honour of Freeman Dyson.

[imhotep]

A very few of you may have heard about Freeman Dyson - The Professor without a PhD. There are many original works associated with Freeman Dyson including the Dyson series. He worked along with Richard Feynman.
J Robert Oppenheimer, then director of the Institute for Advanced Study in Princeton, New Jersey, offered Dyson a lifetime appointment at the Institute.

A Brilliant theoretical physicist and mathematician whose far-fetched ideas for the future verged on the bizarre. He won several medals and prizes.
Here's video of him delivering a guest lecture at the age of 90... He passed away at the age of 96, on Feb 28, 2020.



He was attributed with being able to perform numerical calculations very swiftly. It's said that one day where a number of top scientists and mathematicians got together - someone asked - "Whether an integer number exists, so that you could exactly double by moving its last digit to its front."
For eg, 256 would satisfy this if 625 were its exact double—which it isn’t.

Apparently, in just FIVE seconds Dyson responded, “Of course there is, but the smallest such number has 18 digits.”

Given - Dyson's statement, is it possible to find this number?

 

[topkollek]

food for thought

 

[Stimulus mind]

 

[siri_ayya]

@Wishmalokaya Official කලින් මේවගේ thread එකක උඹව mention කරන්න කියල තිබ්බ නේද? මේකට උත්තරෙත් දීපන් බලන්න

 

[Walter White]

TFS !!!

 

[imhotep]

food for thought

You don't need advanced mathematics. Just year 3 or 4 stuff only... How long addition works.

 

[kinkon]

බම්ප්

 

[Wishmalokaya Official]

A very few of you may have heard about Freeman Dyson - The Professor without a PhD. There are many original works associated with Freeman Dyson including the Dyson series. He worked along with Richard Feynman.
J Robert Oppenheimer, then director of the Institute for Advanced Study in Princeton, New Jersey, offered Dyson a lifetime appointment at the Institute.

A Brilliant theoretical physicist and mathematician whose far-fetched ideas for the future verged on the bizarre. He won several medals and prizes.
Here's video of him delivering a guest lecture at the age of 90... He passed away at the age of 96, on Feb 28, 2020.



He was attributed with being able to perform numerical calculations very swiftly. It's said that one day where a number of top scientists and mathematicians got together - someone asked - "Whether an integer number exists, so that you could exactly double by moving its last digit to its front."
For eg, 256 would satisfy this if 625 were its exact double—which it isn’t.

Apparently, in just FIVE seconds Dyson responded, “Of course there is, but the smallest such number has 18 digits.”

Given - Dyson's statement, is it possible to find this number?

I'm replying before watching or doing a Google search, but I have a hunch that finding any such number would be much easier (and computationally less resource intensive) doing it the other way round, i.e. Finding a number where moving first digit to the end exactly halves it.

Algorithm is quite simple. I'll try to run a Numpy code and check it.

 

[imhotep]

I'm replying before watching or doing a Google search, but I have a hunch that finding any such number would be much easier (and computationally less resource intensive) doing it the other way round, i.e. Finding a number where moving first digit to the end exactly halves it.

Algorithm is quite simple. I'll try to run a Numpy code and check it.

Actually it does not require any computer or an algorithm. Either basic addition or simple multiplication by 2 is all that's required with a bit of clever thinking.

PS: We know it has 18 digits. The genius of Dyson was to deduce the smallest number will have 18 digits. He was a master of number theory.

 

[hasithayad]

Actually it does not require any computer or an algorithm. Either basic addition or simple multiplication by 2 is all that's required with a bit of clever thinking.

PS: We know it has 18 digits. The genius of Dyson was to deduce the smallest number will have 18 digits. He was a master of number theory.

105263157894736842

 

[imhotep]

The above answer by @hasithayad is correct...

We know that the number contains 18 digits. Let's say it is abcdefghijklmnopqr. Then do the addition to itself -and the answer should be rabcdefghijklmnopq

abcdefghijklmnopqr
+abcdefghijklmnopqr
------------------------
rabcdefghijklmnopq

Obviously we know a = 1 (we look for the smallest number and a cannot be zero)
Then we know r = 2 (or at least 2 if there's no carry). Lets assume r = 2

So replacing r by 2 this becomes...
abcdefghijklmnopq2
+abcdefghijklmnopq2
-------------------------
2abcdefghijklmnopq

Now we know q = 4... replace all q by 4

abcdefghijklmnop42
+abcdefghijklmnop42
-------------------------
2abcdefghijklmnop4

So p = 8 and then continue the same operations to fill up the rest. Remember to take the carry over.

You will arrive at the answer 105263157894736842