[VA] Short & Sweet Math Challenge #25 "San Valentin's Special: Weird Math"

[Valentin Albillo]

 
Hi, all !   Happy San Valentín 2021 to all of you !  
 
Almost a year has elapsed since my previous S&SMC, exactly 6 years 6 since I registered in this new-style forum, and today it's San Valentin's Day so here you are, a brand-new Short & Sweet Math Challenge #25  "San Valentin's Special: Weird Math", once again intended to give you a chance to dust off your chosen HP calculator and actually use it, while also flexing your HP-programming muscle *and* your HP-assisted sleuthing abilities (*NOT* your Google Search proficiency).

Try all 6 Concoctions 6 below and see what you can do about them !

Rules:

• Any HP calc of your choice may be used (physical or virtual) but I suggest a Minimum Recommended Model (MRM) for each Concoction, which is the simplest model I deem capable of solving it more or less comfortably. Lesser models might, but not comfortably.
   
• Using anything other than a physical or virtual HP calculator is strictly disallowed, thus no Wolfram Alpha/Mathematica/Matlab/Pari, VBA, Excel. Pascal, C/C#/C++, Java, Python, Lua, etc., please go elsewhere for that. The idea here is that you actually use your HP calc, not your Lua on a laptop, so write your code in a language originally supported on some HP calc (RPN, RPL, 71B BASIC, Hppl, etc).
   
• Googling for the solutions is über-lame and by doing so you're admitting to yourself than you just can't cope and have to cheat to try and save face  .

Concoction the First: Weird limit
[MRM: HP-11C and up]

Many HP calculators have a function that returns a (pseudo-)random number uniformly distributed between 0 (inclusive) and 1 (exclusive), such as RND in the HP-11C, HP-15C, HP-71B and many other models, and RAN in the HP-42S.

Now get your calc and conduct a test where you generate a couple of random numbers and add them up, then if the sum is less than 1 you continue generating and adding up more random numbers one at a time until the sum eventually exceeds 1, while keeping count of just how many random numbers did you generate.

For instance, suppose your first test generates 0.87 and 0.65 and their sum is 1.52, which is greater than 1 already so the test is over and 2 random numbers were generated in all. Now you conduct another test and the sequence of random numbers is 0.21, 0.07, 0.16, 0.35, 0.19 and finally 0.58, which makes the sum (1.56) exceed 1, so the test is over and we had to generate 6 random numbers in all.


The Challenge:

Write a program that simulates the process for a given number of tests and outputs the average count of random numbers generated per test, and then (the sleuthing part) use the program to help you answer these questions:

• a.  What do you think is, in the limit, the average count of generated random numbers for their sum to exceed 1 ? Can you recognize what the exact value would be ?
• b.  What would the average count be for the sum to exceed 2 ? To exceed 2.021 ? To exceed Pi ?

Surprising, isn't it ? But there's more surprises: now get a suitably fast HP model (physical or virtual) and conduct a sizable number of tests (say ~100,000) to find the average count for sums exceeding 3, 4, 5 and 5+1/6 (no kidding, try it) on the one hand, and for sums exceeding 10, 15 and 20 on the other, and analyze the results you get, in order to answer these additional questions:

• c.  What do you think is the asymptotic expression for the average count needed to exceed large values of the sum ?
• d.  Can you explain the constant component of said expression ?

Use a seed of 1 for the random number generator at the very start of your program (for instance, RANDOMIZE 1 on the HP-71B and 1, SEED on the HP-42S) and give your answers accurate to at least 2-3 digits. Please do not use/post any theoretical formulas to get the results for now, do it empirically by just generating and using actual random numbers.

I'll give my original solutions for both the HP-42S and the HP-71B, as well as my comments on the results.


Concoction the Second: Weird Sum
[MRM: HP-11C and up]

Consider the following infinite Albillo sum:



where, other than 2021, the coefficients are the prime numbers p_k in order: p₁ = 2, p₂ = 3, p₃ = 5, p₄ = 7, ...

Note: Observe that for k=1 in the sum above, the product in the numerator is the empty product, thus equal to 1 by definition.


The Challenge:

Write a program to compute and output the sum as accurately as possible, and then (the sleuthing part) use your HP calc (perhaps conduct some experiments) to try and attempt to answer this question: What's so weird about this sum ? (BTW, forget about Googling for it because I concocted it myself and it's nowhere else to be found AFAIK.)

I'll give my original solution for the HP-71B, as well as my comments on the result.


Concoction the Third: Weird Integral
[MRM: HP-15C and up]

Consider the following definite Albillo integral:



where Γ is the Gamma function, ln is the natural logarithm (i.e., base e) and φ is the Golden Ratio = (1+ √5)/2.


The Challenge:

Use your HP calc to compute (either manually or writing a program to do it) and output the value of the definite integral as accurately as possible, and then (the sleuthing part) use your HP calc (perhaps conduct some experiments) to try and attempt to answer this nagging question: What's so weird about this integral ?. (Again, forget about Googling for it because I concocted it myself and it's nowhere else to be found either.)

I'll give my original solution for the HP-71B, as well as my comments on the result.


Concoction the Fourth: Weird Graph
[MRM: HP-PRIME and other graphing models]

Consider the following polynomial in two real variables x, y:

      P(x, y) = 9 x⁸ + 9 y⁸ + 36 x² y⁶ + 54 x⁴ y⁴ + 36 x⁶ y² - 100 x⁶ - 4 y⁶ - 108 x² y⁴ - 204 x⁴ y² + 182 x⁴ - 10 y⁴ - 84 x² y² - 100 x² - 4 y² + 9


The Challenge:

Use your graphing calc (remember: no Wolfram Alpha, etc), either by writing and running some program or manually (but then succintly specify the operations performed) to accurately plot the resulting 2D graph for P(x, y) = 0, and somewhat describe what you see in the graph you get, giving also relevant parameters (say ranges for x and y, or maybe things like centers or focii or radii or asymptotes or intersections or zeros/poles, whatever.

If you think it might help, you may also attempt to factorize the polynomial, but in any case the main question is: What's so weird about this graph ?

I wont post code or manual operations as I don't own any graphing calculators but I'll give the resultant graphic you should get, as well as extensive commentary.


Concoction the Fifth: Weird Primes
[MRM: any fast physical or virtual HP calc]

In Milos Forman's 1984 "Amadeus" film (winner of 8 Academy Awards {aka Oscars}, including Best Picture) Salieri comments on the perfection of Mozart's music:

      "Displace one note and there would be diminishment, displace one phrase and the structure would fall."

Now let's bring that observation to the realm of prime numbers and consider a prime number so 'Perfectly Prime' (a PP for short, pronounced "Pepe") that changing any single digit would diminish its primeness by turning it into a composite number. Note: We're talking about base-10 digits here.


The Challenge:

Write a program (the faster & shorter, the better) for your HP calc to compute: (a) the 5 smallest PP, (b) the first PP greater than 500 million, (c) the first PP greater than 777,777,777 and only for very fast programs/devices, the second PP greater than 666,666,666.

I'll give my original solution for the HP-71B, as well as my comments on the results.


Concoction the Sixth: Weird Year
[MRM: HP-11C and up]

Note: All that follows is stated utterly tongue-in-cheek, not to be taken seriously at all. No offence whatsoever is meant to anyone who's been negatively affected during 2020.

Unless you've been hiding under a rock last year, surely you're sorely aware that 2020 was a catastrophic year and many of you might wonder why did it came that way. I know I did, and being fully convinced that this Universe of ours is a mathematical entity subject to mathematical rules, I have been analyzing the matter exhaustively using my trusty HP calculators and have finally succeeded in unraveling the mystery !! At long last, I now know the reason why 2020 was a catastrophic year and of course the reason is of a mathematical nature, as expected.

To wit, the reason is that 2020 shares a very striking numeric property with many other catastrophic years such as, e.g.: the year 662 (the Damghan earthquake killed 40,000 people), the year 458 (the Antioch earthquake killed 80,000), the year 1348 (the Black Death plague, which killed up to 200 million, was at its apogee), the year 1556 (the Shaanxi earthquake killed 830,000) or the year 1849 (the Great Irish Famine killed ~1,500,000), to name but a few.

That can't be a mere coincidence !  Moreover and just in case this wasn't evidence enough, the number 666 (the infamous Number of the Beast of apocalyptic fame) also shares that very property as well.


The Challenge:

Use your trusty HP calculator to assist you in your sleuthing to try and discover what simple but highly remarkable (striking, in fact) numeric property all the aforementioned numbers have in common, and then write a program to find out and output a listing of all years between AD 4 and AD 5000 (both included) which have this property (hint: less than 100). Of course the listing should include all mentioned past years as well as future years thus predicted to be potentially catastrophic, up to that limit.

The questions are: (a) How many years will be listed in the output ? (b) What will be the next predicted potentially catastrophic year after 2020 ?, and (c) Should we be concerned ?

As an additional hint to help finding the remarkable shared property, remember Occam's Razor: the property can be unambiguously stated by saying that the year's number is a "(five words)".

I'll give my solution, two short programs (6 and 7 lines resp.) for the HP-71B which produce the listing and also accept a given year in range and demonstrate^* whether it has the required numeric property (thus, if it indeed was/might be catastrophic) or not.

• * E.g.: For property "The year's number is a factorial" and year 720 you would output "720 = 1x2x3x4x5x6" demonstrating the property, while for year 721 you would ouput "721 = not a factorial".

And last but certainly not least, a couple' important caveats:

• Please do NOT include CODE panels in your replies to this thread, as it makes it difficult for me to generate the online PDF document which will include the whole thread. I expect you'll kindly comply with this requirement but otherwise you'll risk your carefully crafted code appearing truncated or not at all in the final PDF and thus being irretrievably lost from the online document and making your posting it moot. With no CODE panels, to prevent solutions posted before yours spoiling your fun, I advise to simply avoid reading any of them before posting your own. Thank you.
   
• Designing, testing and formatting these Challenges and their solutions takes and awful lot of time and effort. Hence, if you do enjoy them and would like to see more posted in the future, consider participating or commenting on them so that I get feedback of your appreciation.

I'll post my original solutions in a week or so, for you to have enough time (and no excuses) to try them all. That is, if you've got what it takes ...

That's all. Enjoy ! ... and that's an order !

V.