Metals of the millions

[Just another guyn he/him]

Is this correct:

 

Every shard can combine with the regular 16 metals to form 16 new alomantic alloys plus every shard combination because harmony and harmonium then there are a total 1,048,576 possible alomantic metals and powers if all sixteen shards and their combinations are used. This includes the known god metals.

this is based on the facts that: It has been confirmed by Brandon that what ruin is to atium, Honor is to shardblades

Based on the formula

File:Https://www.mathsisfun.com/combinatorics/images/cn.svg

Where N is the number of shards and are can equal any whole number between 1 and 16 (sixteen shards) plus the additional non god metals as one more combination, the following determines the maximum number of allomantic alloys

(65535 + 1) * 16 = 1,048,576

 

 

support for this theory 

http://www.theoryland.com/intvmain.php?i=692#28

 

Edited January 19, 2017 by Just another guyn

[Oversleep]

Do you think there is a difference between lerasium alloy of malatium and atium alloy of lerasium-gold alloy?

Anyway, I think you may want to take a look at that.

[The One Who Connects he/him]

6 minutes ago, Just another guyn said:

Is this correct:

Every shard can combine with the regular 16 metals to form 16 new Allomantic alloys plus every shard combination because harmony and harmonium

Not exactly

Quote

If Odium were lured to Scadrial, would his physical body turn into a burnable metal? If so, could Harmony create an Odium-metal legion of Mistings to consume and burn it?

Brandon Sanderson

The difficulty here is, again, one of Identity. People born on Scadrial have an Identity tied to it and its magic. Odium would have to do certain things to make them able to use a magic he fuels. He has done these things on Roshar, so it's not impossible for him to manage it on Scadrial.

I suppose if you want to go purely theoretical, then there would be quite a few. I'll let someone else check the math, since I am either misunderstanding what you wrote, or the equation in the image, but I feel that the number should be lower than that.

[Just another guyn he/him]

14 hours ago, Oversleep said:

Do you think there is a difference between lerasium alloy of malatium and atium alloy of lerasium-gold alloy?

Anyway, I think you may want to take a look at that.

I do, since the metals themselves do different things

[Spoolofwhool]

2 hours ago, Just another guyn said:

I do, since the metals themselves do different things

Except once you alloy the metals you're still ending up with anew alloy of the three metals. Alloying appears to be path independent so having them do different things is to have different metal ratios, which isn't a thing allomantically.

[The One Who Connects he/him]

Following the logic that differing ratios of the same metals are not new metals, there are a limited number of combinations. Imagine that the 16 Shards are A-P (first 16 letters of the alphabet)

Quote

16 shards, 16 god metals. 16
2 metal combos with Metal A:15 Now there's no A, so there is only 15 pairs to be made of the remaining 15, and so forth down the line till there is only 1 combination (O&P) 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15 = 120
The math continues the same way for 3 God metal combo's but starting at 14 on down. But once you reach 3 or more combo's, it gets more difficult. There are 14 AB, 13 AC and 13 BC, 12 AD & 12 BD & 12 CD, and on. 3 God metal combo's have a total of 560, but it spirals down from there.
4 metal combo's: [Some attempt at doing permutation, with combination math, attempting to correct for it not being combinations.. It was bad]  and finally, 16 metal combo's: 1

16 + 120 + 560 + [wrong] + 1 = 2536 god metals and solely god metal alloys. Each of those metals can alloy with any of the 16 normal metals. 2536*16 = 40576 total Allomantic alloys, nowhere near the Million you reached.

Adding in Harmonium and Harmonium-God metal alloys would add in 1 god metal, 16 pairs, 120 triples, [more flawed math] and 1 17 combo. Those new numbers combined = 697. 2536 + 697 = 3233. Repeating the alloy with any of the 16 normal metals from earlier, that makes the new total 51,728 Allomantic Metals and Alloys.

If anyone wants to fact-check my math, feel free. I added the Harmonium numbers while writing this post, so those may have been a tad rushed. As for the OP, I am curious how you did your math, since I am at a loss to figure out how you reached such a high number.

edit: I did some sort of franken-math, and clearly didn't know what I was doing at the time. If only mobile had the strikethrough text button, then it would look cleaner. Edit: fixed.

Edited January 20, 2017 by The One Who Connects
All the strikethroughs

[Just another guyn he/him]

18 hours ago, Oversleep said:

Do you think there is a difference between lerasium alloy of malatium and atium alloy of lerasium-gold alloy?

Anyway, I think you may want to take a look at that.

18 hours ago, The One Who Connects said:

Not exactly

I suppose if you want to go purely theoretical, then there would be quite a few. I'll let someone else check the math, since I am either misunderstanding what you wrote, or the equation in the image, but I feel that the number should be lower than that.

See http://www.theoryland.com/intvmain.php?i=1181

[The One Who Connects he/him]

3 minutes ago, Just another guyn said:

See http://www.theoryland.com/intvmain.php?i=1181

After having perused that Tour, I am forced to assume that you mean Question 31.

Quote

If an Allomancer worldhopper really wanted to hack the magic system and knew what they were doing, could they get their hands on some Tanavast-ium or Rayse-ium or Egdli-um? Basically make god metals from the other shards?

Brandon Sanderson

Yes.

It is still a manner of hacking, but on the opposite side of the coin compared to the WoB I posted. In the earlier one, Odium would have to do the workarounds to form a usable God Metal, while in yours, the Allomancer is the one hacking the system. Nothing is really disputed by having both, but it is interesting to note that you can hack Shardic Investiture into a specific magic system.

[Pagerunner he/him]

23 minutes ago, The One Who Connects said:

Following the logic that differing ratios of the same metals are not new metals, there are a limited number of combinations. Not counting Harmonium at all yet, I am getting a number in the low 40,000's. Imagine that the 16 Shards are A-P (first 16 letters of the alphabet)

16 + 120 + 560 + 455 + 364 + 306 + 220 + 165 + 120 + 84 + 56 + 35 + 20 + 10 + 4 + 1 = 2536 god metals and solely god metal alloys.
Each of those metals can alloy with any of the 16 normal metals. 2536*16 = 40576 total Allomantic alloys, nowhere near the Million you reached.

Adding in Harmonium and Harmonium-God metal alloys would add in 1 god metal, 16 pairs, 120 triples, 105 quads, 91 5's, 78 6's, 66 7's, 55 8's, 45 9's, 36 10's, 28 11's, 21 12's, 15 13's, 10 14's, 6 15's, 3 16's ans 1 17 combo. Those new numbers combined = 697. 2536 + 697 = 3233. Repeating the alloy with any of the 16 normal metals from earlier, that makes the new total 51,728 Allomantic Metals and Alloys.

If anyone wants to fact-check my math, feel free. I added the Harmonium numbers while writing this post, so those may have been a tad rushed. As for the OP, I am curious how you did your math, since I am at a loss to figure out how you reached such a high number.

Can you go into more detail for your math? OP's math checked out when I did it. # of combinations ascended until I reached 8 Shards, of which I saw 12,870 combinations. (Just take a look at your proposal that there are only four 15-Shard combinations. But, we know there must be 16; each different alloy is excluding a single Shard, so you can define a 15-combo as a 'Ruin-absent Adonalsium,' or something like that. The pattern persists across all combinations that add up to 16; 0 and 16 reflect, 1 and 15 reflect, 6 and 10 reflect, 8 reflects with itself. (Ew.) 

Here's the detailed explanation for my math:

• For a double combination, you've got to choose two metals, one at a time. First one, you choose out of 16 options. Second one, you choose out of 15 options (because the first one is taken), for 240 options (16*15). But, then you need to account that the order isn't actually important; AB is the same as BA. So, you have to look at the different ways you can arrange AB, which is 2, so we need to divide our total by 2 for 120 options. (So far, so good, with your math. The technical terms are 'permutations' for order-specific, and 'combinations' for not-order-specific.)
• For a triple combination, you make three choices. 16*15*14. But, a combo ABC can be arranged in 6 ways (ABC, ACB, BAC, BCA, CAB, CBA), so we need to divide by 6 to eliminate all the many duplicates. That leaves 560 options. (Still good.)
• For quadruple, 16*15*14*13. Arrangements: ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC... well, I'm not gonna type them all out, but there are 24 in total. That math works out to 1820. (You're off by a factor of 4.)
• For 5, I got 4638, where you were lower by a factor of 12.
• The general form, for combining any number N Shards out of 16, is: (16!) / [(16-N)! * N!)], which is the equation that was provided originally, I believe. (The link isn't working for me anymore.) When we add up from N=0 (the 'normal metal' state) to N=16 (Adonalsiumiuim), we get 65,536 possibilities for each of the 16 Allomantic metals. (Which is why it's multiplied by 16 to get the >1E6 number.)

Harmonium is already accounted for, supposing that harmonium-atium and harmonium-lerasium are not a thing.

However, there is also a missing 'null state' that isn't shown: god metals that aren't alloyed with a regular metal. OP should have multiplied by 17 instead of 16 at the end, and then subtracted one for the no god metal and no regular metal.

[The One Who Connects he/him]

1 minute ago, Pagerunner said:

The technical terms are 'permutations' for order-specific, and 'combinations' for not-order-specific.)

Thank you for this, the distinction about "order-specificity" was never pointed out in High School, I just used whichever one seemed to fit in my head.

4 minutes ago, Pagerunner said:

Can you go into more detail for your math? [..] But, we know there must be 16; each different alloy is excluding a single Shard, so you can define a 15-combo as a 'Ruin-absent Adonalsium,' or something like that.

I think that concept is where I screwed up, but at a far earlier number. What I did was with 16 letters, there are 14 combinations that have AB in them. Since I've already done ABC, there would be 1 less triple with AC in it, and so forth down the line. Same for BC, BD, and the rest of the letters. I used that same logic all the way up to 7 before I noticed a pattern and just did numbers from there. Looking at your math explanation, I basically did addition rather than multiplication, resulting in vastly lower numbers as time went on.

7 minutes ago, Pagerunner said:

The general form, for combining any number N Shards out of 16, is: (16!) / [(16-N)! * N!)], which is the equation that was provided originally, I believe. (The link isn't working for me anymore.) When we add up from N=0 (the 'normal metal' state) to N=16 (Adonalsiumiuim), we get 65,536 possibilities for each of the 16 Allomantic metals. (Which is why it's multiplied by 16 to get the >1E6 number.)

The link never worked, you had to click it and then cut out everything before the http in the middle of the URL. I couldn't get the math to work because for some strange reason, I mixed up which was N and R, thus getting me negative numbers half of the time.

I'm doing the formula in excel now, and it's working out a lot better. I don't know why I couldn't get it to work earlier.

[Dunkum he/him]

2 hours ago, Pagerunner said:

However, there is also a missing 'null state' that isn't shown: god metals that aren't alloyed with a regular metal. OP should have multiplied by 17 instead of 16 at the end, and then subtracted one for the no god metal and no regular metal.

Just double checking the math, but I think this is correct.  (65536*17) -1=1114111 according to google (side note, I like how that number looks a lot better :P).

for my own math: the number of possible combinations of k items out of a group of n total items is n choose k, sum as k goes from 0 to n of n choose k is 2^n (had to google this, but it is a direct corollary of the binomial theorem), so total possible number of god metal combinations (including the no godmetal case) is 2^16 or 65536. then as you say all the alloys plus the pure godmetal case, minus the no metal case.

[Oversleep]

Assuming there is only one alloy of given metals (so no difference between lerasium alloy of malatium and atium alloy of lerasium-gold alloy):
The simplest way to count is it like this:

Godmetal #1: it's in the alloy or not (2 options)
Godmetal #2: it's in the alloy or not (2 options)
...
Godmetal #16: it's in the alloy or not (2 options)

so 2*2*2...*2 = 2^16. That's what we can get out of 16 godmetals (including nothing). But then we also may want to include 16 basic metals (or we may not want, so multiply by 17). So all in all, it's 17* 2^16. Or 1 114 112. But wait, this can mean no godmetal and no basic metal. That's nothing. We don't want that. Substract one. 1 114 111.

Mind you, this is without harmonium. But with harmonium we just multiply the first number by 2 (harmonium or no harmonium), then substract "no metal", so 2 228 223.
This does not include "no harmonium alloys of atium or/and lerasium" thing.

Edited January 20, 2017 by Oversleep

[Oversleep]

1 minute ago, Dunkum said:

yea, that number doesn't account for questions like is there actually a difference between harmonium(shouldn't we call this sazedium or maybe sazium) and an atium/lerasium alloy (for the record, I assume those would be the same thing).  in which case the actual possible number is much higher: i think it maxes out around (2^65536)*17 -1 if we both assume that harmonium is separate from atium/lerasium alloy and that you could have a distinct effect from a harmonium/atium or harmonium/lerasium or harmoniumatium/lerasium alloy.  maybe higher if we allow for splinters to be their own thing....

Ummm nope.

It acounts for difference between harmonium and lerasium/atium alloy. It doesn't include the suggested rule for not alloying harmonium with atium and/or lerasium.

[Dunkum he/him]

14 minutes ago, Oversleep said:

Ummm nope.

It acounts for difference between harmonium and lerasium/atium alloy. It doesn't include the suggested rule for not alloying harmonium with atium and/or lerasium.

when I said "that number", I was referring to my original one, 1114111, though I see where my phrasing is a bit bad; should have said "my number".  the one you suggested clearly does allow for a harmonium/atium etc combinations.  my later number 2^65536 etc. allows for any combination of godmetals made from any combination of shards.

[Djarskublar he/him]

I don't remember the math well enough to agree or disagree with any of these answers, except for @The One Who Connects's answer... sorry. I think the next important question to ask is how this would affect probabilities of becoming and Allomancer. Assuming that all of these alloys are made to be like FE atium, then they should have mistings for them once the Shard is invested. There may be more of these god metals that are like Lerasium and can be burned by anybody, but I don't think that is a good assumption at the moment. The question then lies in what the effect is on the proportion of people who gain Allomantic abilities, and then also how it affects your probability of having specific powers. Would it be a straight even probability, so that any of the powers is literally ~1/1,000,000? Would it be related to the Connections you had to specific Shards at birth? Would it be more along the line of which Shard has the most power invested (and therefor, probably the base 16 and/or atium would be the most common, since there are, presumably, no Lerasium mistings)? Or would it be some weird Realmatic interaction that deals with some random cosmic factors surrounding your birth that would determine which you would be. Interesting lines of thought that we don't have the resources to answer effectively, unfortunately.

[Just another guyn he/him]

After further reading, there is also the possibility of more base metals then just 16. Just thought I would mention that.

[Oversleep]

2 hours ago, Just another guyn said:

After further reading, there is also the possibility of more base metals then just 16. Just thought I would mention that.

I doubt it very much. Brandon has always talked about 16 base metals and everything reinforces this further (for example, Preservation himself talks about 16 as the number of metals). So, do you mind linking a source for that?

Edited April 10, 2017 by Oversleep

[+Extesian he/him]

3 hours ago, Oversleep said:

I doubt it very much. Brandon has always talked about 16 base metals and everything reinforces this further (for example, Preservation himself talks about 16 as the number of metals). So, do you mind linking a source for that?

Yeah we have this and I've never seen it contradicted

Quote

QUESTION

For the space Mistborn are we going to have more elements?

BRANDON SANDERSON

The base 16 are basically it. But the interactions between them and things. And there is one more metal, there is harmonium, so you will hear about that later on.

Source

The only other ones will be god metals and their alloys which, as Brandon days below, aren't really metals.

Quote

QUESTION

I actually have a weird question. From the Mistborn series it says there are 16 Allomantic metals but then you go into The Alloy of Law and the 16 are listed there, minus the atium and another one, so are there really 18 metals?

BRANDON SANDERSON

Well, you see those two were not really metals. Those were pieces, fragments, of a god

 

[Pagerunner he/him]

4 hours ago, Oversleep said:

I doubt it very much. Brandon has always talked about 16 base metals and everything reinforces this further (for example, Preservation himself talks about 16 as the number of metals). So, do you mind linking a source for that?

24 minutes ago, Extesian said:

Yeah we have this and I've never seen it contradicted

The only other ones will be god metals and their alloys which, as Brandon days below, aren't really metals.

 

I believe this is the WoB in question:

Quote

Q:(seems to ask how the metals for mistborn were chosen, random , or was there something behind it)I was thinking, was there (...) random or was something behind it?

A: I wanted the metals that had an alloy, that was commonly used and is easily accessible to people in a pre-industrial society

Q:(?) [there is something about “copper” ?]^[a](When they go and discover more contents of the periodic table, is there a chance they’ll discover (?...))

A:There is a chance, yes.

It's not very clear, but it can be read that some trace metals may have an Allomantic effect. It's possible that the question-asker hadn't read MB Era 2, and wasn't aware of nicrosil or chromium.

[Oversleep]

9 hours ago, Pagerunner said:

I believe this is the WoB in question:

Quote

Q:(seems to ask how the metals for mistborn were chosen, random , or was there something behind it)I was thinking, was there (...) random or was something behind it?

A: I wanted the metals that had an alloy, that was commonly used and is easily accessible to people in a pre-industrial society

Q:(?) [there is something about “copper” ?]^[a](When they go and discover more contents of the periodic table, is there a chance they’ll discover (?...))

A:There is a chance, yes.

It's not very clear, but it can be read that some trace metals may have an Allomantic effect. It's possible that the question-asker hadn't read MB Era 2, and wasn't aware of nicrosil or chromium.

I will ask around but I doubt I will find the person who asked that. (BTW, the comment about copper is mine but I may have misheard)

[Just another guyn he/him]

 Just asked Brandon about it at a book signing, and he said that there are over a million theoretical possible metals, however there are not actually that many usable ones. I asked about that being due to some exploding on contact with water like harmonium, and he partially rafoed me, and said that harmonium would be very difficult to burn as it "reacts violently with nearly everything"

[Dunkum he/him]

40 minutes ago, Oversleep said:

Assuming there is only one alloy of given metals (so no difference between lerasium alloy of malatium and atium alloy of lerasium-gold alloy):
The simplest way to count is it like this:

Godmetal #1: it's in the alloy or not (2 options)
Godmetal #2: it's in the alloy or not (2 options)
...
Godmetal #16: it's in the alloy or not (2 options)

so 2*2*2...*2 = 2^16. That's what we can get out of 16 godmetals (including nothing). But then we also may want to include 16 basic metals (or we may not want, so multiply by 17). So all in all, it's 17* 2^16. Or 1 114 112. But wait, this can mean no godmetal and no basic metal. That's nothing. We don't want that. Substract one. 1 114 111.

Mind you, this is without harmonium. But with harmonium we just multiply the first number by 2 (harmonium or no harmonium), then substract "no metal", so 2 228 223.
This does not include "no harmonium alloys of atium or/and lerasium" thing.

yea, that number doesn't account for questions like is there actually a difference between harmonium(shouldn't we call this sazedium or maybe sazium) and an atium/lerasium alloy (for the record, I assume those would be the same thing).  in which case the actual possible number is much higher: i think it maxes out around (2^65536)*17 -1 if we both assume that harmonium is separate from atium/lerasium alloy and that you could have a distinct effect from a harmonium/atium or harmonium/lerasium or harmoniumatium/lerasium alloy.  maybe higher if we allow for splinters to be their own thing....