# What Imprisonment sentence will immortal citizen soldier Guilt receive?

On planet Immortal, everyone lives forever but essential resources (including food and supplies) on the planet are limited. Ultimatopian raiding swords are the most effective weapon available to loot nearby planets.

Ultimatopian raiding swords are created from a limited supply of ultimatopium. For every 17 swords recycled one more can be created. Each sword can be used in battle once and exactly once but can always be recycled again at the same 17/1 ratio.

To maximize the utility of planetary resources, the planet requires immortal citizens to recycle diligently. The penalty for failure to recycle a ultimatopian sword after battle is a mandatory 10 years imprisonment for each ultimatopian sword that can never be created as a result of the failure to recycle (assuming perfect recycling compliance from every other soldier).

All known ultimatopium has now been exhausted and the only remaining Ultimatopian swords totaling 83,521 have just returned from battle to the sword recycling center.

Assume:

• 1 sword is destroyed by immortal citizen soldier Guilt after a later battle in protest for an innocent mortal citizen who was carelessly killed on Planet Weakling during a raid.
• No other immortal citizen soldiers are ever imprisoned because all other available swords are turned in for recycling.

1. What is the maximum imprisonment immortal citizen soldier Guilt could receive?
2. What is the minimum imprisonment immortal citizen soldier Guilt could receive?
• I hope he at least feels guilty about it.
– COTO
Jul 1, 2016 at 17:18

Thanks to the answer Freddy Flares has already given (Part 2) and reminding me of an important thing I didn't take into account, i was able to get:

Part 1:

There will be 4912 swords that can be recycled in the second cycle, since all the swords for the first cycle are already collected so he breaks one after that.

4912 - 16 = 4896 Swords to be recycled (16 stored for later)
4896 / 17 = 288 Swords made
288 - 16 = 272 Swords to be recycled (16 stored for later)
272 / 17 = 16 Swords made
16 + 16 = 32 Swords left 32 / 17 = 1 Sword made
leaving 15 swords, ( 1 unmade sword ) laying around.
Resulting in 10 years of imprisonment

Part 2:

83521 used swords are recycled ( 83521/17 ) = 4913 Swords
4913 used swords are recycled ( 4913/17 ) = 289 Swords
289 used swords are recycled ( 289/17 ) = 17 Swords
17 used swords are recycled ( 17/17 ) = 1 Sword

And if it's the last sword that he breaks it doesn't matter because there were no more swords that could've been recycled along with it.

Maximum 10 years, Minimum 0 years

• Re part 1: Wouldn't there be 16 used swords left over from the first recycling batch which can be combined to make 289 swords in the second batch? Jun 30, 2016 at 6:46
• Actually, you're right, damn, didn't think about the fact that there would be 16 remaining because they couldn't make them into a sword. Jun 30, 2016 at 6:55

Soldier Guilt gets away scot-free.
83521 used swords are recycled into 83521/17 = 4913 swords.
4913 used swords are recycled into 4913/17 = 289 swords.
289 used swords are recycled into 289/17 = 17 swords.
17 used swords are recycled into 17/17 = 1 sword.
This last sword is destroyed by Soldier Guilt which was never going be used to recycle anyway (because there are no other swords left to combine with it) and hence no sentence is applicable.

Maximum sentence is

10 years

because

He destroys a sword, that leaves the planet with 83520, with that you can create 4912 swords ... BUT that leaves you with 16 unrecycled swords to use at the next recycle round. Thus, with those (4912+16) swords, you create 289 swords, leaving you with 15 unrecycled swords. With these, you craft 17 new swords, leaving 15 unrecycled swords ... and at last, you can craft a last sword. You'll then have only 16 swords in total and won't be able to create a new one.

So ..

with a sword destroyed, you have 4912 +289+17+1 = 5219 swords created, and
with no sword destroyed, you have 4913+289+17+1 = 5220 swords created.
Leaving us with a difference of 1, thus, 10 years.

But of course, as Benjo said ..

if the sword destroyed is the last, then, no more sword could have been created, so .. Guilt's free.

The key is to count the number of swords which "can never be created". 83,521 swords have just returned from battle (let's call it the battle of 17e4). We know that Guilt destroys his sword after "a later battle". So, we know that after recycling, the next battle (the battle of 17e3) will have at maximum 4,913 swords.

If Guilt destroys his sword after the battle of 17e3, 17e2, or 17e1, there will be used swords available for recycling when all is said and done.

Which means that his (maximum) sentence will be:

10 years (if swords were not able to carry over between battles, it could have been 30 years) The battle of 17e2 will be one sword short (288 instead of 289) with 16 "used" swords in storage. We will have 304 used swords for recycling. (More than the 289 we would have had) The battle of 17e1 will have a full 17 swords, with 15 used swords in storage. We still have one sword for 17e0, but can do nothing with the remaining used swords. In total, 1 sword will not have been created as a result of Guilt's actions, resulting in a 10 year term.

However, if the "later battle" is 17e0 he'll get off much lighter:

No jail time at all! Since Guilt's sword was the last one anyway, there was no way to recycle it into another sword.

• Except, when the sword is destroyed post 17e4 you have the 16 unused swords left lying around from post 17e3 when you get to 17e2.. presumably the recycling center can hold on to them if they aren't used immediately!
– Arth
Jun 30, 2016 at 16:45
• Ah :) I suppose "can always be recycled again" should have made that clear. Jun 30, 2016 at 16:52

83521 used swords are recycled ( 83521/17 ) = 4913 Swords.

4913 is 17 cubed, so if Guilt destroys one of these swords, one less sword can be created in the next generation: 4912/17 = 288 (with 16 left over) instead of 289 (4913/17)

However for the next recycling, one of the 16 swords left over can substitute for the sword not created, so 289 swords are recycled to create 17 swords (with 15 still left over). I.e. the same number of swords are created in this recycling generation.

The 17 swords are then recycled to create 1 more sword, with 15 still left over. No more swords can be created.

Thus only one less sword can be created, so the sentence is 10 years.

As previous answers stated, if he destroys the last sword he gets 0 years because the sword couldn't have been recycled.

Thus the maximum sentence is 10 years, and the minimum sentence is 0 years.

In both cases...

He walks away scot free. Why? A lawyer is technical in the terminology, it says:

[...] the only remaining ultimatopian swords totaling 83,521 have just returned from battle to the sword recycling center.

More specifically:

It says they were just returned. Given there is only one sword left over, he could argue that no swords were destroyed as a result. But he has another get out clause...

Assume 1 sword is destroyed by immortal citizen soldier Guilt after a later battle

It occurs in a later battle, AFTER the swords have been recycled... which means soldier Guilt gets off... scot free!

In maximum case -

10 years as he destroyed one sword.

In minimum case -

He Will Walk away for free as he is not the immortal citizen who recycles swords.(I assume the immortal citizen involved in recycling process are only charged, not the immortal citizen soldiers). Even if we calculate 83,521/17 = 4913 so there aren't any other swords wasted at all.

Roughly:

80,000 years imprisonment

Reasoning:

That one sword was neccessary for combining with swords in a latter found source of the recycled material (after all, nobody said other planets didnt have the ore). That combined sword then went on to be nessecasry for conquering yet another planet (the battle was that slim). Without it the other planet was lost and never used as a source. Since the current planet must have enough for at least 8000 swords, we can assume a later planet would have the potential for 8,000. Therefore, his sentence is 80,000 years. Not because the sword was needed in the actual combining of 8000 swords but because it prevented the aquizition of the material for 8000 swords.