Xmas 2021 Puzzle(1)

[L D[AR2P#0.0]]

Find 3 palindromic numbers that, when added together, make 85709
(Numbers are in Base 10)


Answers by pm please.

 

[L D[AR2P#0.0]]

A few solutions in already.

 

[pasbra]

It'a not easy...

https://fr.wikipedia.org/wiki/Nombre_palindrome

 

[PLCnovice61]

here this possible answer, donation to D, over $10000000 tax deductible
https://www.plctalk.net/qanda/forumdisplay.php?f=2

 

[I_Automation]

You forgot to include if you thought of the puzzle on a train and what color the cat was.

 

[L D[AR2P#0.0]]

Puzzle inspired by the following paper:


https://arxiv.org/pdf/1602.06208.pdf


There are lots of solutions for 85709

 

[drbitboy]

There are lots of solutions for 85709

I found 382 excluding palindromic numbers with even numbers of digits

 

[Mispeld]

All-in -- one through five-digit palindromes -- I found 957 solutions.

There is an Excel workbook named PalPuzzle.xlsm in the file download section, Misc folder. It is not necessary to enable macros to view the 957 palindrome sums for 85709 in the Solution worksheet. Macros are required to try other numbers via the Puzzle worksheet.

 

[drbitboy]

There is an Excel workbook named PalPuzzle.xlsm in the file download section, Misc folder.

Nice!


And d'Oh! 0 is a palindromic number, so my 382 claim for odd-number-of-digits palindromic triplets is one short!

 

[drbitboy]

...I found 957 solutions...

I duplicated @Mispeld's results (thanks for the correction!): the same 957 solutions; interestingly, they come from 956(!) possible palindromic decimal numbers that are less than 85709

Source is in Downloads: xmas21_puzzle_1.zip under Misc; 17 active lines of Python; runs in ~6s or less on my laptop.

 

[AustralIan]

9.9 is I guess palendromic

 

[drbitboy]

9.9 is I guess palendromic

Dude, that is seriously Outside the box. Sweet! Did you find any solutions using that yet? I think there are only 5 possibilties for the single integral value, per 5-digits target sum.

 

[Mispeld]

9.9 is I guess palendromic

Palindrome, yes in the textual sense, but not numeric per the definition in the paper linked in post #6.

Nevertheless, stretching the definition to include a decimal point (i.e., #.#, ##.##, ###.###) does not affect the result for 85709. Still just 957 solutions.

Some smaller test values tend to get many more hits, such as the value 110 goes from 22 integer solutions all the way to 617 when floating point palindromes are considered. The following clip shows a few examples.



Setting aside the unconventional (non-integer) palindromes, I ran some tests to understand the efficiency of the algorithm running in the uploaded Excel workbook. The attached tables and associated plots show the results for different input values, 8571, 85709, 857090, and 8570900. In this case the time increases linearly, indicating O on the magnitude of the input value.

More interesting is what happens when the number of addends changes. This is the number of palindromes added together to reach the target value. The puzzle defines this as three, and is the focus of the proof in the post #6 paper.

You can see in the Log-Y chart that result time increases exponentially with the number of addends O(10^n). Not desirable, but also not surprising with the brute-force structure that almost certainly repeatedly, and needlessly, re-evaluates the same partial results. Might be something to look at before going back to the mundane, real-world programming.

Numeric Benchmarks:



Graphs:

 

[AustralIan]

Nevertheless, stretching the definition to include a decimal point (i.e., #.#, ##.##, ###.###) does not affect the result for 85709. Still just 957 solutions.

Some smaller test values tend to get many more hits, such as the value 110 goes from 22 integer solutions all the way to 617 when floating point palindromes are considered.

Nice work!

 

[drbitboy]

Sweet.


You inspired me to get my three-addend solution for 85709 down to around 0.05-0.06s, and only 15 active lines.