Inglenook options calculations

[Jeremy Cumberland]

1 hour ago, Colin_McLeod said:

My head hurts.

Why does selecting the three you are not going to use, give a different answer to selecting the five you are going to use?

It doesn't. The answer is 56 either way.

 

In my previous post, I was assuming that the order of the five wagons in the finished train mattered, which is why I used 6,720 possible train combinations and came up with 44,706,816,000 permutations for the puzzle (including functional duplications). If the order of wagons in the finished train does not matter, then there are 56 possible train combinations and 372,556,800 permutations for the puzzle.

 

Both of these big numbers can be divided by six, because the three unused wagons can always have their starting positions swapped with each other, giving 7,451,136,000 and 62,092,800 respectively. The actual number of distinct functional permutations is less than this, but I don't know a way of calculating it.

 

24 minutes ago, SteveyDee68 said:

(note that this calculation only takes into account the rolling stock present on the layout and disregards the distribution of the three "empty slots" in the sidings, as these are not part of the object of the puzzle itself and only serve as manoeuvering space; if you were to factor them in then the number of combinations rises significantly).

I don't know how you can disregard them, unless you always begin with the same number of wagons in each siding, such as always having one of the short sidings empty.

 

On the other hand, with three "spaces", some of the 6,652,800 starting possibilities are functionally equivalent so, for example, the long siding could begin with - -123, -1-23, -12-3, -123-, 1--23, 1-2-3, 1-23- or 123--, and the only difference between these is how far to move the engine.

[SteveyDee68]

38 minutes ago, Jeremy Cumberland said:

I don't know how you can disregard them, unless you always begin with the same number of wagons in each siding, such as always having one of the short sidings empty.

I was simply quoting Adrian Wymann’s page - I’m assuming they are ignored to simplify (?!) the calculations.

 

Steve S

[SteveyDee68]

40 minutes ago, Jeremy Cumberland said:

On the other hand, with three "spaces", some of the 6,652,800 starting possibilities are functionally equivalent so, for example, the long siding could begin with - -123, -1-23, -12-3, -123-, 1--23, 1-2-3, 1-23- or 123--, and the only difference between these is how far to move the engine.

Which is maybe also why spaces are ignored when calculating - you’ve shown 8 variations on that one siding, so if the three wagons were instead nos 2, 3 and 4 then that’s another 8 etc; if nos 1, 3, 4, then another etc etc

 

Much simpler to ignore the spaces! 🤣
 

Steve S

[goldfish]

1 hour ago, Jeremy Cumberland said:

Both of these big numbers can be divided by six, because the three unused wagons can always have their starting positions swapped with each other, giving 7,451,136,000 and 62,092,800 respectively. The actual number of distinct functional permutations is less than this, but I don't know a way of calculating it.

 

Calculating the number of possible starting positions for 8 wagons with 11 possible starting locations includes every possible combination. It is not possible to have duplicates because each starting situation is unique. The 5 wagons selected at the end can have any order, but the starting position is fixed.

[SteveyDee68]

1 hour ago, goldfish said:

The 5 wagons selected at the end can have any order, but the starting position is fixed.

I always thought the point of the Inglenook puzzle was to organise those five selected wagons into a particular order, in which case “the 5 wagons selected at the end can have any order” is incorrect!

 

Unless we are talking mathematical order? (I have this nagging feeling that that is something different to the norm, but I haven’t studied maths beyond O Level … not surprisingly, considering I held the school record for consecutive non-completed maths homework — 67, I thank you! 🫢😆)

 

Steve S

[goldfish]

56 minutes ago, SteveyDee68 said:

I always thought the point of the Inglenook puzzle was to organise those five selected wagons into a particular order, in which case “the 5 wagons selected at the end can have any order” is incorrect!

There appears to be various views about the 5 selected wagons being in a particular order or not. Personally I always go for a specified order because it makes the puzzle that more interesting. The practical difference is that there are 120 times more combinations if the wagons are ordered. Definitely a case where rule #1 applies.

[BoD]

Has anybody considered the mathematics of somebody slipping in a bogie bolster as one of the wagons?

 

Thought not.

[Mol_PMB]

12 minutes ago, BoD said:

Has anybody considered the mathematics of somebody slipping in a bogie bolster as one of the wagons?

 

Thought not.

My MSC Inglenook has a bogie bolster. But importantly, it counts as TWO wagons.

 

[Stubby47]

2 hours ago, BoD said:

Has anybody considered the mathematics of somebody slipping in a bogie bolster as one of the wagons?

 

Thought not.

 

In theory, you can use any 8 items of rolling stock, in any length of siding, as long as you stick to the rule of only 5, 3, 3 items each.

 

So they could be 8 x Mk3 coaches...

[Stubby47]

Everyone, many thanks for your input.

 

I am of the persuasion that the 5 wagons need to be in a specific order, with the other 3 being any order and in any 'spot'.

 

Despite the concensus that there are only 6720 actual permutations,  I still can't see why my original calculation was in correct - I'm sure it is, I just don't know why.

[Thorness]

If you really want your brain scrambled try this:

 

Blackburn2019-InglenookShuntingPuzzles.pdf

 

[Jeremy Cumberland]

I don't know that this subject requires any further discussion, but while I was out walking this afternoon, I did think of a considerable simplification for the permutations of starting positions of the 8 wagons in a set of sidings that can hold 5 wagons, 3 wagons and 3 wagons.

 

The permutations for 8 wagons in 8 positions is 40,320. We have 11 positions, but many of these, where it would result in gaps in the middle of a siding, can be disregarded. In fact, we have only 7 6 possible configurations for the 8 wagons:

- 5 wagons in the long siding, 3 wagons in one of the short sidings

- 5 wagons in the long siding, 2 wagons in one short siding and 1 wagon in the other

- 4 wagons in the long siding, 3 wagons in one short siding and 1 wagon in the other

- 4 wagons in the long siding, 2 wagons in each of the short sidings

- 3 wagons in the long siding, 3 wagons in one short siding and 2 wagons in the other

- 2 wagons in the long siding, 3 wagons in each of the short sidings.

 

This reduces our starting possibilities to a more manageable 282,240 241,920. Note that we do not need to consider the two short sidings as siding A and siding B, and so count 2 wagons in siding A and 1 wagon in siding B separately from 1 wagon in siding A and 2 wagons in siding B, since functionally they are equivalent, and you could, in describing a particular shunt, call the two short sidings "the short siding with two wagons" and "the the short siding with one wagon".

 

If we also consider the make up of the train, we can divide this by six, because whatever the choice of wagons to form the train, the other three wagons can each be swapped with each other. This reduces the total to 47040 40,320 distinct starting possibilities for any given train. It is actually a little less than this, as we can demonstrate by considering the trivial situation of the train already being formed up in the long siding in the correct order. The number 47040 40,320 includes two permutations for this, one where the other three wagons are all in one siding and one where two are in one siding and one is in another. In terms of solving the puzzle, these starting positions are functionally the same. There will be a handful more functionally equivalent permutations, and 47040 40,320 is a small enough number to run through them all in Excel, for example, to weed out the other functional duplicates, should anyone wish to do so.

 

If the order of wagons in the train does not matter, then the functionally distinct starting permutations reduces dramatically. I divided by 6 to make all the non-train wagons the same, and I think that by dividing by 120 will make all the train wagons the same (rather than having wagons 1 to 8, we have 5 train wagons and 3 non-train wagons), which brings the number of starting permutations for any given train down to 392 336 (and once again, a few of these will be functionally equivalent.

 

Bed time.

Edited July 20 by Jeremy Cumberland
It was late and it seems I can't count as far as 6 without making a balls up of it and calling it 7.

[kevinlms]

4 hours ago, Stubby47 said:

 

In theory, you can use any 8 items of rolling stock, in any length of siding, as long as you stick to the rule of only 5, 3, 3 items each.

 

So they could be 8 x Mk3 coaches...

Not really, because HST's tended to have fixed formations, typically with 1st accommodation at one end and 2nd at the other, with catering vehicles inbetween. So the shunting would be BORING.

[SteveyDee68]

@Stubby47 said “In theory, you can use any 8 items of rolling stock”

 

That is true, as long as each individual item conforms to the “standard wagon unit length” that is used to measure the length of sidings. It happens to be that 10’ wheelbase wagons are more compact than say, mk1 coaches or bogie fertiliser wagons.

 

When I first rejoined the hobby, I was really eager to do something modern image having discovered the small container terminal at Irlam on the Manchester Ship Canal. To that end, I started collecting Bachmann container wagons, a Freightliner 66, an 08 And even a class 70. My plan was to build an Inglenook based on the ‘port’, a 3-2-2 configuration, as each “wagon” would actually be a pair of bogie wagons, but each pair treated as a single “wagon unit”

 

My reasoning was that instead of different wagons being selected and ordered, it would be the liveries of the containers themselves! Brilliant, I thought!!

 

Except if you want to rearrange the order of containers on a train, you simply lift them off and rearrange them, not shunt the wagons about… 🙄

 

I really should move on the large wrapping paper box I have of modern image freight stock that I no longer have an immediate use for, but I got everything at such good prices that I am loathe to let them go … just in case! 🫢🙄🤣

 

The maths is really an aside to the real point of an Inglenook, of course - which is that such layouts provide lots of shunting fun!

 

Steve S

 

(Struggling to sleep in the heat!)

Edited July 20 by SteveyDee68

[Stubby47]

3 hours ago, kevinlms said:

Not really, because HST's tended to have fixed formations, typically with 1st accommodation at one end and 2nd at the other, with catering vehicles inbetween. So the shunting would be BORING.

Ah, but someone has to put the coaches into the fixed rake formation....

[goldfish]

9 hours ago, Stubby47 said:

I am of the persuasion that the 5 wagons need to be in a specific order, with the other 3 being any order and in any 'spot'.

 

Despite the concensus that there are only 6720 actual permutations,  I still can't see why my original calculation was in correct - I'm sure it is, I just don't know why.

 

Just why your original calculation was incorrect is difficult to explain, but you start by calculating the number of ways of arranging 5 wagons,  5!  (120). You then replace one of the wagons. To do this you simply multiply by 5, what you should have done at this point is select 5 wagons from 6, 720 ( 6! / 1!,  ). This approach is incorrect because what you are doing is starting out with 5 wagons and calculating the number of arranging them, and then calculating the number of ways ways of replacing a wagon. This not what is required, what is required is to start with 8 wagons and then calculate the number of ways of selecting 5 wagons from 8. You were starting from the wrong premise, does that make sense?

 

Where you did err was to fail to specify the starting and finishing positions wagons, these make a huge difference to the possible permutations. Personally I take the starting position of a run of the puzzle as being the final position of the previous run of the puzzle, and regard the final order and positions of all the wagons to be important so that the starting conditions are always the same. In particular no siding can be empty can be empty at the start of the puzzle.

Given those restrictions there are 40320 ( 8! ) potential starting positions, 6720 ( 8! / 3! ) ways of arranging the 5 selected wagons. So there are 270,950,400 possible ways of running the puzzle, ignoring the arrangement of the other 3 wagons.

 

Just complicate thing my shunting plank is an inglenook with and additional siding to accommodate another 3 wagons, so it is a 5-3-3-3 arrangement with 11 wagons. I'll spare you the calculations for that...

[Stubby47]

I think I understand, but not sure why the starting position is important, when the object of the puzzle is to sort 5 of the 8 into a given order.

Yes, where a wagon is originally might affect the difficulty of an individual puzzle, but not the result.

[SteveyDee68]

3 hours ago, goldfish said:

Just to complicate things my shunting plank is an inglenook with an additional siding to accommodate another 3 wagons, so it is a 5-3-3-3 arrangement with 11 wagons. I'll spare you the calculations for that...

Now, that sounds fun! Of course, as it is still only selecting 5 wagons, it wouldn’t be much longer in overall length than the ‘original’ Inglenook, as the headshunt would remain 3 wagons + loco.

 

Of course, your “long” siding could actually be longer than 5 wagons, but have perhaps “move blocker” wagons permanently parked at the end to reduce the available slots to 5 wagons.

 

Or perhaps there’s just the extra room to park a brake van at the end to block the spaces - as the “finished” consist is usually presented as being about to be hauled off scene via the headshunt, it would be in the right place visually. The fact it never does depart…

 

Oh dear … I am starting to get the urge to build that obligatory Inglenook that every modeller should do! 🤣

 

Looking forward to seeing what @Stubby47 builds!

 

Steve S

[2E Sub Shed]

10 hours ago, kevinlms said:

Not really, because HST's tended to have fixed formations, typically with 1st accommodation at one end and 2nd at the other, with catering vehicles inbetween. So the shunting would be BORING.

Having been in a HST set with a 1st class coach replacing a 2nd class coach halfway down the formation, and numerous times odd vehicles were removed / exchanged for servicing, sometimes creating a short set, there may be some possibilities.

[Stubby47]

4 hours ago, 2E Sub Shed said:

Having been in a HST set with a 1st class coach replacing a 2nd class coach halfway down the formation, and numerous times odd vehicles were removed / exchanged for servicing, sometimes creating a short set, there may be some possibilities.

Exactly!!

[kevinlms]

12 hours ago, 2E Sub Shed said:

Having been in a HST set with a 1st class coach replacing a 2nd class coach halfway down the formation, and numerous times odd vehicles were removed / exchanged for servicing, sometimes creating a short set, there may be some possibilities.

I've made maybe 6 trips on an HST and had one like that, but I don't remember now where this coach was in the formation. In fact I had an entire 1st class coach to myself! I was travelling with a 1st class ticket, so I could sit wherever I liked. Anyway this coach had a label at the first class end to say that it was for 2nd class passengers. So no self-respecting 1st class passenger would use it!

There was no such sign at the 2nd class end of the train, so passengers didn't use it, because they didn't know that they could use it!

 

Hence I had it to myself! The really funny part was when the ticket inspector came through, he saw my 1st class ticket and pointed out that this particular coach was for 2nd class. I just told him that I knew that and was happy sitting in it anyway - after all it was still a 1st class vehicle.

 

He did go through the rest of the train, but no one came through from the 2nd class direction, so obviously he wasn't telling anyone that they could!

 

Besides, I did say that the USUAL formation was 1st class one end and the other end 2nd class of an HST, so that is still a valid argument.

 

I don't normally travel first class, but at the time overseas visitors could buy Brit Rail passes before leaving home. They were a bargain and there was virtually no difference in price, for 2nd vs. 1st class, I don't remember how much, maybe 5% - so I treated myself!!!!!!!!!!!

[CloggyDog]

Someone has already published a mathematical thesis on Inglenooks

 

https://arxiv.org/abs/1810.07970

Edited July 21 by CloggyDog

[JohnR]

2 hours ago, CloggyDog said:

Someone has already published a mathematical thesis on Inglenooks

 

https://arxiv.org/abs/1810.07970

 

Already referred to upthread.

[Nick C]

On 19/07/2024 at 22:36, Stubby47 said:

 

In theory, you can use any 8 items of rolling stock, in any length of siding, as long as you stick to the rule of only 5, 3, 3 items each.

 

So they could be 8 x Mk3 coaches...

I've not got room for it at the moment, but I've had an idea for a while of building an inglenook set in the corner of a carriage works.

 

I mentioned this to @brightspark at a show while chatting about Express Daisy, and he suggested an EMU depot for a serious inglenook - imagine shunting 8 4-car units round...

[03060]

3 hours ago, Nick C said:

I've not got room for it at the moment, but I've had an idea for a while of building an inglenook set in the corner of a carriage works.

 

I mentioned this to @brightspark at a show while chatting about Express Daisy, and he suggested an EMU depot for a serious inglenook - imagine shunting 8 4-car units round...

 

I've had similar thoughts regarding both a DMU depot or a parcels area within a station both inspired by the Bradford Interchange and Hammerton Street area. Currently trying to build an Inglenook into an IOW scheme which I hope to 'unveil' once a bit more progress has been made.

 

Regards,

Ian.