Bulgarian Solitaire

First popularized by Martin Gardner, Bulgarian Solitaire is a simple yet mathematically fascinating card game which has kept clever minds busy ever since it was first introduced in the early 1980’s.
Sources:
Bulgarian solitaire. (2014, December 15). In Wikipedia, The Free Encyclopedia. Retrieved from en.wikipedia.o...
Drensky, V. (2010). The Bulgarian solitaire and the mathematics around it. Math and Education in Math, Proc. of the 44-th Spring Conf. of the Union of Bulgar. Mathematicians, SOK-Kamchia, April 2-6, 2015, 79-91. arXiv:1503.00885v1 [math.CO]. www.math.bas.bg... PK/tom 2015/pdf/079-091.pdf.
(arxiv.org/pdf/...)
Öhman, E. (2012). Multiplayer Bulgarian solitaire [PDF]. Retrieved from static1.square....
Partition (number theory). (2017, June 20). In Wikipedia, The Free Encyclopedia. Retrieved from en.wikipedia.o....
Music: ”Life of Riley" Kevin MacLeod (incompetech.com)
Licensed under Creative Commons: By Attribution 3.0 License
creativecommons...
Sound effects obtained from www.zapsplat.com

[@alkankondo89]

I love (A) how the field of math has SOOOO many of these subcategories that spring from real-world applications and (B) that, of these subcategories, this channel always seems to present the ones that no other channel (e.g. Numberphile) is doing. This channel was well worth the subscription!

[@6u6u7o]

alkankondo89 You bet! Numberphile has become quite dull over the years, not like this channel which has stayed relevant and interesting.

[@bananabananae]

3blue1brown is another good channel. You can check it out.

[@alkankondo89]

Ah, yes! I am actually already subscribed to 3Blue1Brown. The visuals he provides in his videos are amazing and super-effective! Also, "Mathologer" and "MindYourDecisions" provide pretty good math content. I am subscribed to both. Also, "blackpenredpen" is a great channel if you want to really sharpen your problem-solving skills (e.g. in Calculus).

[@MrMaxHolm]

You are making fantastic content. Thank you very much and keep up the good work!

[@Michaelonyoutub]

would the smallest 3 cycle be 17? if 1 being filled, 0 being a hole, the final row could be 110000, 1010000, or 100100 which each have different cycles.

[@hamminglin1019]

I think you're right! I thought it was 12 before but two of the combinations would have been the same. And since 17 is 2 greater than the next smallest triangular number it must be the smallest with 3 distinct cycles

[@paulsaves7816]

12/ 13 seems to works the same (2/5 holes or 2/5 blocks : 2 cycles), as for 17 and 19 (4/6 holes or 4/6 blocks: 3 cycles). for 18 (3holes and 3 blocks), it seems to be -3 cycles too.-

[@petemagnuson7357]

For 18, I think there are four cycles: 111000, 110100, 101100, 101010.

[@paulsaves7816]

That's true, 8 is composed,on the upper step, with 2 blocks and 2 holes and it's the 2 cycles minimum ; like 18 (with 3 blocks and 3 holes) which is the 4 cycles minimum. I've counted 10 cycles for 32 (whith 4 blocks and 4 holes) but I don't know if it's the 10 cycles minimum.

[@completeandunabridged.4606]

This is great, keep up the good work. Sir.

[@joxfon]

Since we need three different cycles, we need three different gap shifts. The number of gaps is constant, so the smallest gap sequences are CGG, CGCG, CGCCG, where G is gap and C is card. The extra C to the left is to make GG distinct to GCCG. 5 slots in diagonal are a 5 level triangle, wich has 15 slots. Finally, 15 slots - 2 spaces = 13 cards. So 13 is the smallest number of cards with three distinct cycles.

[@joxfon]

Oh, allright, now I see it. Thank you :)

[@6u6u7o]

I believe that the answer is 12. 2 blocks on the top and 3 holes make 3 possible configurations with different cycles :)

[@bluewildflower12]

Wow the graphics are exceptional! Do you know who did them?

[@TippingPointMath]

My student Libby.

[@ratandmonkey2982]

Nice work Libby !

[@bluewildflower12]

Oh my gosh, a student did this? That's impressive!

[@odenpetersen6028]

This would look cool with a big number of cards

[@Yotam1703]

I'm guessing 7? Because it's the lowest number that's both 3 less than a triangular(10) AND not a triangular itself (3,6)?

[@canteatpi]

if 7 would be the lowest number with 3 cycles, it would also be the smallest with 2, which is 8, so 7 can not be the answer. Also 7 has only one cycle of period 4

[@Yotam1703]

Alain Hoffmann thank you for answering! I realized that after looking through the comments.

[@canteatpi]

No problem, I like your kind reply

[@BigDBrian]

love this

[@sirgermaine]

No views?

[@aajjeee]

my guess is 13