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. Subsets are inputted in the order they were discovered using the Instruction Table on the next page.

Variables

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

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 multiple subsets of r = 1 exist (j > 1), Subset N merges them (r = j1).	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 as a rounded down whole number AND its decimal form. Subset N Value(s): Index A1 / CPSn = SnV	Count up to the SnVth (whole) combination of Subset N. See Counting Subset Values on Page 3. Save Subset N as part of the final answer (if r = j1, unmerge 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 two values of SnV: It’s whole number form AND decimal form. (decimal(SnV) - whole(SnV)) × IPC = Index A1	Index A1 replaces Index A2

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. This also applies if the rest of the subsets are subsets of r = 1. 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 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 is given the values [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).

If SnV is too large a number to count quickly, refer to Appendix A on the next page for a strategy that resembles C# code.

Appendix A

int v = the # of already set elements;
int x = 0; int y = 0; int w = 0;
for (int i = 0; until Subset N is solved; i + 1) {
x = value in cell[column r, row v];
y + x;
if (y > SnV) {
SnV = SnV - (y - x);
While avoiding already set elements:
SubsetN[i] = value in cell[ row w, the leftmost column ];
v = v + SubsetN[i] + 1
r - 1; w = 0; }
else if (y = SnV) {
for (int i2 = 1; until Subset N is solved; i2++;) {
SubsetN[i] = value in cell[ row w + i2, the leftmost column );
i + 1; }
w + 1; }

If SnV becomes low enough, you may go back to counting Subset N incrementally.

	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