On the Subject of 27,644,437

A scientific calculator is recommended! Please use a scientific calculator!
Use a scientific calculator!

The module contains:

• A circle of 13 white LEDs.
  • Ordered 0-12 clockwise from the top.
• A lower screen cycling through sets 0-12.
• An upper screen displaying a Base-36 number. (100000-ZZZZZZ)

A partition is a particular combination of subsets (groups) that can be made with a set of n elements. There are 27,644,437 possible partitions in a set of 13 elements, the LEDs being the elements, and the Base-36 number being the ID to the partition. Clicking the LEDs will toggle their presence in the selected subset, those being the letters A-M.

Partitions can be displayed in two ways:

• Set Partition: Shows which specific elements are in each subset:
•         [[0], [2], [7], [1,8], [3,12], [5,10,11], [4,6,9]]
• Integer Partition: Shows how many elements are in each subset:
•         [1, 1, 1, 2, 2, 3, 3]

Locate and submit the partition pointed to by Base-36 number in order to solve the module. Use the Instruction Table on the next page. The order you input subsets does not matter, and the module will never strike.

Variables

The following is a list of variables that will appear regularly throughout the instructions. Each will always equal an integer.

n	Element amount. Starts at 13, n - r1 for each subset found.
r	Subset length r1 = 1 = [x]    r2 = 2 = [x,x]
j	Amount of a specific subset. j1 = 2 = [x], [x]    j2 = 3 = [x,x], [x,x], [x,x]

Instruction Table

1.	Convert the base of the module’s serial number, becoming the “Base Index”. See Converting Bases on Page 3.	Perform the following operation: Base Index % 27,644,437	Locate the cell in Appendix B on page 5 that matches with the Base Index.
2.	Take the PTF and IPC of the cell found in Appendix B, then plug into the formula.	Base Index - ((PTF + 1) - IPC) = Index A1
3.	Locate Subset N, the subset with the smallest r that has not been affixed a value. *If the smallest subset has an r equal to 1, Subset N is r = j of (r = 1).	Use the current IPC and the SnC of the current Subset N. IPC (Integer Partition Combos): n! / ( r1! × r2! × r3!… × j1! × j2! × j3!… ) SnC (Subset N Combos): n! / ( r! × (n - r)! )	CPSn (Combos per single value of Subset N): IPC / SnC = CPSn
4.	Find SnV. Save it both as a rounded-down integer, AND its non-integer form if present. SnV (Subset N Value(s)): Index A1 / CPSn = SnV	Count up to the SnVth (int.) combination of Subset N. See Counting Subset Values on Page 3. Save Subset N as part of the final answer. If r = j of (r = 1), split elements.	Remove Subset N’s r from the integer partition.
5.	Recalculate IPC with Subset N removed. n = n - (Subset N's r) j1 = j1 - 1, j != 0	Separate SnV into its integer AND non-integer form, then plug into the formula. (non_int(SnV) - int(SnV)) × IPC = Index A2	Index A2 replaces Index A1

6.	If Index A2 = 0, continue from here. If not, repeat Steps 3-5 until it does.	If one subset is left, any unset elements may be added to it. If multiple subsets are unset, use the first available combination for each subset in order of whichever subset would be Subset N next.	Once the correct partition is inputted, the module will solve.

Converting Bases

To convert from Base-36 to Base-10, take each letter’s alphanumeric value and add 9 to it. Then starting from the leftmost number, multiply each number by 36ⁿ. n in this case starts at 5, and decreases incrementally by one for the next number. Finally, add each product.

ABZ129 ⇒ 10, 11, 35, 1, 2, 9
(10 × 36⁵) + (11 × 36⁴) + (35 × 36³) + (1 × 36²) + (2 × 36¹) + (9 × 36⁰)
(10 × 60,466,176) + (11 × 1,679,616) + (35 × 46,656) + (1 × 1,296) + (2 × 36) + (9 × 1)
= 624,771,873

Counting Subset Values

For a subset of r = 2, its combinations are ordered/indexed like so. Subsets with a higher r are ordered similarly:
SnV = 0123…
[0, 1], [0, 2], [0, 3], [0, 4], …, [0, 9], [0, 10], [0, 11], [0, 12],
SnV = 121314…
[1, 2], [1, 3], [1, 4], …, [1, 9], [1, 10], [1, 11], [1, 12],
SnV = 2324…
[2, 3], [2, 4], …, [2, 10], [2, 11], [2, 12], …

If a subset of r = 2 were to be given the elements [4, 9], later subsets can not use either of the elements, as they are already fixed to a subset. Select the next available element(s). Subsets will always list elements from least to greatest.

If SnV is too large a number to count quickly, refer to Appendix A.

Appendix A

Select the cell at column = r, row = v. At the start, v = current # of set elements.
Keep a list of every set and unset element.

Take a variable s = 0. Add the selected cell's value to s, then every cell below it, stopping at a cell once the addition of its value to s makes it greater than SnV. Don't add the value of the stopped-at cell.
Take the number of steps it took to reach the stopped-at cell from the selected cell, as a variable e.
Then, select the eth element in your list of unset elements in order, zero-indexed. Set the selected element to Subset N.

If r > 1, repeat this process for the rest of Subset N given that:

• SnV = SnV - s
• r = r - 1
• v = v + e + 1
• unset elements with values less than the current eth element are ignored when solving the rest of Subset N.

	1	2	3	4	5	6	7	8	9	10	11	12	13
0	1	12	66	220	495	792	924	792	495	220	66	12	1
1	1	11	55	165	330	462	462	330	165	55	11	1
2	1	10	45	120	210	252	210	120	45	10	1
3	1	9	36	84	126	126	84	36	9	1
4	1	8	28	56	70	56	28	8	1
5	1	7	21	35	35	21	7	1
6	1	6	15	20	15	6	1
7	1	5	10	10	5	1
8	1	4	6	4	1
9	1	3	3	1
10	1	2	1
11	1	1
12	1

Appendix B

In reading order, the list displays every possible integer partition that can be made in a list of 13 elements.

PTF = Partitions Thus Far (the index for each int. partition)
IPC = Integer Partition Combos (# of set partitions in an int. partition)

The Base Index matches with a cell if BOTH:

• Base Index is greater than the previous cell’s PTF
• Base Index is less than or equal to the selected cell’s PTF

Shade	~Amount	If a PTF surpasses a multiple(s) of one million partitions, the cell will be shaded for the sake of convenience. Format of cells: r0, r1, r2, ... [IPC] PTF
LightDark Gray	1,000,000
DarkLight Gray	2,000,000
BlackWhite	3,000,000

13 [1] 0	12, 1 [13] 13	11, 2 [78] 91	11, 1, 1 [78] 169	10, 3 [286] 455
10, 2, 1 [858] 1313	10, 1, 1, 1 [286] 1599	9, 4 [715] 2314	9, 3, 1 [2860] 5174	9, 2, 2 [2145] 7319
9, 2, 1, 1 [4290] 11609	9, 1, 1, 1, 1 [715] 12324	8, 5 [1287] 13611	8, 4, 1 [6435] 20046	8, 3, 2 [12870] 32916
8, 3, 1, 1 [12870] 45786	8, 2, 2, 1 [19305] 65091	8, 2, 1, 1, 1 [12870] 77961	8, 1, 1, 1, 1, 1 [1287] 79248	7, 6 [1716] 80964
7, 5, 1 [10296] 91261	7, 4, 2 [25740] 117001	7, 4, 1, 1 [25740] 142741	7, 3, 3 [17160] 159901	7, 3, 2, 1 [102960] 262861

7, 3, 1, 1, 1 [34320] 297180	7, 2, 2, 2 [25740] 322920	7, 2, 2, 1, 1 [77220] 400140	7, 2, 1, 1, 1, 1 [25740] 425880	7, 1, 1, 1, 1, 1, 1 [1716] 427596
6, 6, 1 [6006] 433602	6, 5, 2 [36036] 469638	6, 5, 1, 1 [36036] 505674	6, 4, 3 [60060] 565734	6, 4, 2, 1 [180180] 745914
6, 4, 1, 1, 1 [60060] 805974	6, 3, 3, 1 [120120] 926094	6, 3, 2, 2 [180180] 1106274	6, 3, 2, 1, 1 [360360] 1466634	6, 3, 1, 1, 1, 1 [60060] 1526694
6, 2, 2, 2, 1 [180180] 1706874	6, 2, 2, 1, 1, 1 [180180] 1887054	6, 2, 1, 1, 1, 1, 1 [36036] 1923090	6, 1, 1, 1, 1, 1, 1, 1 [1716] 1924806	5, 5, 3 [36036] 1960842
5, 5, 2, 1 [108108] 2068950	5, 5, 1, 1, 1 [36036] 2104986	5, 4, 4 [45045] 2150031	5, 4, 3, 1 [360360] 2510391	5, 4, 2, 2 [270270] 2780661
5, 4, 2, 1, 1 [540540] 3321201	5, 4, 1, 1, 1, 1 [90090] 3411291	5, 3, 3, 2 [360360] 3771651	5, 3, 3, 1, 1 [360360] 4132011	5, 3, 2, 2, 1 [1081080] 5213091
5, 3, 2, 1, 1, 1 [720720] 5933811	5, 3, 1, 1, 1, 1, 1 [72072] 6005883	5, 2, 2, 2, 2 [135135] 6141018	5, 2, 2, 2, 1, 1 [540540] 6681558	5, 2, 2, 1, 1, 1, 1 [270270] 6951828
5, 2, 1, 1, 1, 1, 1, 1 [36036] 6987864	5, 1, 1, 1, 1, 1, 1, 1, 1 [1287] 6989151	4, 4, 4, 1 [75075] 7064226	4, 4, 3, 2 [450450] 7514676	4, 4, 3, 1, 1 [450450] 7965126
4, 4, 2, 2, 1 [675675] 8640801	4, 4, 2, 1, 1, 1 [450450] 9091251	4, 4, 1, 1, 1, 1, 1 [45045] 9136296	4, 3, 3, 3 [200200] 9336496	4, 3, 3, 2, 1 [1801800] 11138296
4, 3, 3, 1, 1, 1 [600600] 11738896	4, 3, 2, 2, 2 [900900] 12639796	4, 3, 2, 2, 1, 1 [2702700] 15342496	4, 3, 2, 1, 1, 1, 1 [900900] 16243396	4, 3, 1, 1, 1, 1, 1, 1 [60060] 16303456

4, 2, 2, 2, 2, 1 [675675] 16979131	4, 2, 2, 2, 1, 1, 1 [900900] 17880031	4, 2, 2, 1, 1, 1, 1, 1 [270270] 18150301	4, 2, 1, 1, 1, 1, 1, 1, 1 [25740] 18176041	4, 1, 1, 1, 1, 1, 1, 1, 1, 1 [715] 18176756
3, 3, 3, 3, 1 [200200] 18376956	3, 3, 3, 2, 2 [600600] 18977556	3, 3, 3, 2, 1, 1 [1201200] 20178756	3, 3, 3, 1, 1, 1, 1 [200200] 20378956	3, 3, 2, 2, 2, 1 [1801800] 22180756
3, 3, 2, 2, 1, 1, 1 [1801800] 23982556	3, 3, 2, 1, 1, 1, 1, 1 [360360] 24342916	3, 3, 1, 1, 1, 1, 1, 1, 1 [17160] 24360076	3, 2, 2, 2, 2, 2 [270270] 24630346	3, 2, 2, 2, 2, 1, 1 [1351350] 25981696
3, 2, 2, 2, 1, 1, 1, 1 [900900] 26882596	3, 2, 2, 1, 1, 1, 1, 1, 1 [180180] 27062776	3, 2, 1, 1, 1, 1, 1, 1, 1, 1 [12870] 27075646	3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 [286] 27075932	2, 2, 2, 2, 2, 2, 1 [135135] 27211067
2, 2, 2, 2, 2, 1, 1, 1 [270270] 27481337	2, 2, 2, 2, 1, 1, 1, 1, 1 [135135] 27616472	2, 2, 2, 1, 1, 1, 1, 1, 1, 1 [25740] 27642212	2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 [2145] 27644357	2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 [78] 27644435
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 [1] 27644436