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RPL Micro-Challenge: Christmas in July - Joe Horn - 07-17-2017 02:26 PM
Everybody knows the "12 Days of Christmas" Song (Partridge in a Pear Tree etc). How many gifts has my true love given to me altogether by the end of Day 3? Well, the sum of today's 6 gifts (3+2+1) plus yesterday's 3 gifts (2+1) plus the first day's 1 gift comes to 10 gifts in all. So the Total Gifts through Day 3 = TG(3) = 10.
What is the smallest possible (byte-count) User RPL program that calculates TG(x)?
To check your program, here are a few sample inputs and outputs:
TG(3)=10
TG(5)=35
TG(12)=364
TG(365)=8171255
N.B. This is called a micro-challenge instead of a mini-challenge because a REALLY SMALL solution exists. If it's obvious to you, please don't post it right away; let others try to find it first. "We only appreciate the answer if we struggle with the question." -- Ignatius O'Brien, O.P.
RE: RPL Micro-Challenge: Christmas in July - Werner - 07-17-2017 02:53 PM
I still know the formula by heart, and its shorter form.
4 commands, 2 of which are single digit numbers ;-)
That would then be 20 bytes including the Rpl markers << >>
Cheers, Werner
RE: RPL Micro-Challenge: Christmas in July - Gilles59 - 07-17-2017 03:45 PM
(07-17-2017 02:53 PM)Werner Wrote: I still know the formula by heart, and its shorter form.
4 commands, 2 of which are single digit numbers ;-)
Cheers, Werner
I think I got the idea, but more than 4 commands ....
RE: RPL Micro-Challenge: Christmas in July - Gerson W. Barbosa - 07-17-2017 04:09 PM
32.5 bytes. Only first attempt, though.
RE: RPL Micro-Challenge: Christmas in July - Gerson W. Barbosa - 07-17-2017 04:32 PM
More like it: 27.5 bytes.
PS: Even more like it: 20 bytes.
RE: RPL Micro-Challenge: Christmas in July - Jim Horn - 07-17-2017 04:54 PM
Great question, Joe! Finite differences quickly gave me the formula; factoring it gave a simplified form (also 4 operations and two single digit constants). Several attempts keep giving me 9 step solutions. Do you have a shorter one? (RPN form used; LBL and RTN not counted)
RE: RPL Micro-Challenge: Christmas in July - Gilles59 - 07-17-2017 05:58 PM
(07-17-2017 04:32 PM)Gerson W. Barbosa Wrote: More like it: 27.5 bytes.
PS: Even more like it: 20 bytes.
I was happy with my 38 bytes solution but i have to search another way
RE: RPL Micro-Challenge: Christmas in July - Gerald H - 07-17-2017 06:52 PM
32.5 & stuck.
RE: RPL Micro-Challenge: Christmas in July - Gerson W. Barbosa - 07-17-2017 08:17 PM
(07-17-2017 06:52 PM)Gerald H Wrote: 32.5 & stuck.
Just factor the expression you already have. This might give you an insight towards a shorter solution.
RE: RPL Micro-Challenge: Christmas in July - Gilles59 - 07-17-2017 08:31 PM
(07-17-2017 08:17 PM)Gerson W. Barbosa Wrote:(07-17-2017 06:52 PM)Gerald H Wrote: 32.5 & stuck.
Just factor the expression you already have. This might give you an insight towards a shorter solution.
Yes but I cannot imagine how you arrive at 20 bytes ... Did you have a division in your expression ?
[EDIT] Eureka! I get it!
RE: RPL Micro-Challenge: Christmas in July - Gerson W. Barbosa - 07-17-2017 09:33 PM
(07-17-2017 08:31 PM)Gilles59 Wrote: [EDIT] Eureka! I get it!
Congratulations!
Apparently the 20-byte record cannot be broken.
Gerson.
RE: RPL Micro-Challenge: Christmas in July - Joe Horn - 07-18-2017 12:04 AM
(07-17-2017 04:54 PM)Jim Horn Wrote: Great question, Joe! Finite differences quickly gave me the formula; factoring it gave a simplified form (also 4 operations and two single digit constants). Several attempts keep giving me 9 step solutions. Do you have a shorter one? (RPN form used; LBL and RTN not counted)
This is an RPL challenge, but please feel free to implement it for the RPN machine of your choice.
I wonder whether the polynomial form or the shorter form would be faster on a machine that doesn't have the secret command that's needed (such as the bare-bones HP-41).(07-17-2017 09:33 PM)Gerson W. Barbosa Wrote: Apparently the 20-byte record cannot be broken.
Yes, I'm sure that we're all arriving at the same 20-byte solution.
Final hint to those still hunting: the formula can be expressed this way, which should remind you of a shorter way of expressing it: \(\frac { x(x+1)(x+2) }{ 1\cdot 2\cdot 3 } \)
RE: RPL Micro-Challenge: Christmas in July - Dave Britten - 07-18-2017 02:17 AM
7 steps (not including LBL and RTN) is the shortest I can get it so far on the 20S.
RE: RPL Micro-Challenge: Christmas in July - Paul Dale - 07-18-2017 04:24 AM
Four steps on the 34S.
Pauli
RE: RPL Micro-Challenge: Christmas in July - Jim Horn - 07-18-2017 06:31 PM
Ausgezeichnet! I had "6" in the denominator so didn't recognize the now-obvious simplification as per Paul's note. Many thanks, Joe - a clever insight on your part!
RE: RPL Micro-Challenge: Christmas in July - Gerson W. Barbosa - 07-18-2017 09:07 PM
(07-17-2017 05:58 PM)Gilles59 Wrote:(07-17-2017 04:32 PM)Gerson W. Barbosa Wrote: More like it: 27.5 bytes.
PS: Even more like it: 20 bytes.
I was happy with my 38 bytes solution but i have to search another way
Now that you've found it I should tell you my next program yesterday was 38.5 bytes long.
An exotic memory-consuming solution:
« x x x IDN →DIAG x x x x »
I know there's a better equivalent of the two middle commands, but I don't remember which.
Don't anyone be misled by this one. Joe's optimum solution is simply « y y y y » :-)
RE: RPL Micro-Challenge: Christmas in July - Gerald H - 07-19-2017 09:41 AM
All well & good saying you have a solution with so many bytes but eventually you lay your cards on the table.
Below a 32.5 Byte solution to the problem:
Code:
« DUPDUP 3 + * 2 +
* 6 /
»RE: RPL Micro-Challenge: Christmas in July - Gilles59 - 07-19-2017 11:50 AM
A 38 bytes solution
Code:
<< [ 1 3 2 0 ] SWAP PEVAL 6 / >>RE: RPL Micro-Challenge: Christmas in July - Joe Horn - 07-19-2017 12:56 PM
(07-19-2017 09:41 AM)Gerald H Wrote: All well & good saying you have a solution with so many bytes but eventually you lay your cards on the table.
Ok, we are far enough down the page to reveal the 20-byte solution:
« 2 + 3 COMB »
Why it works: The formula for TG(x) is \(\frac { x(x+1)(x+2) }{ 1\cdot 2\cdot 3 }\), but that looks very similar to the formula for COMB(x,3) which is \(\frac { x(x-1)(x-2) }{ 1\cdot 2\cdot 3 }\). All you have to do to turn the latter into the former is add 2 to x, and voila, TG(x) = COMB(x+2,3).
This is an example of program optimization being obtained not by code optimization but by math optimization.
EDIT: I just noticed something cool. If you run the program above on an input of 'X' (undefined), and then EVAL the resulting mess, you get this:
\[\frac { { X }^{ 3 }+3{ X }^{ 2 }+2X }{ 6 }\]
Now press FACTOR (or COLLECT) and see this:
\[\frac { X\cdot (X+1)\cdot (X+2) }{ 3\cdot 2 }\]
It almost does all the thinking for you.
RE: RPL Micro-Challenge: Christmas in July - Dave Britten - 07-19-2017 01:42 PM
Oh, duh, I was doing COMB(x+2, x-1), completely forgetting about the symmetric nature of combinations.