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One of the possibilities on the Surge table is the following :

11-12 Roll a d10. Your height changes by a number of inches equal to the roll. If the roll is odd, you shrink. If the roll is even, you grow.

Considering that getting this effect 10 times results in a summed height change of +5 inches (-1+2-3+4-5+6-7+8-9+10), the more a Wild Mage can get this effect, the more likely he is to grow taller. My question would be to know what would be the average daily growth of a Wild Mage, considering the following :

  1. As confirmed in this question, it is possible for a Wild Magic Sorcerer to cause Wild Magic Surges with rituals. This means that, with a surge-friendly DM and the use of "Tides of Chaos" between each ritual to guarantee a surge, such a surge may happen each 10 minutes (and 6 seconds). If I'm not mistaken, an adventurer has up to 8 hours of available time each day to do what he wants (including casting rituals), at least in Adventurers League Organized Play. That lets you fit 47 surges in the allowed 8h span, leaving 5 minutes and 18 seconds unused (or 53 turns). These can be used to cast non-ritual sorcerer spells of 1st+ level, assuming that the Wild Mage will convert all of his 2nd, 3rd, 4th and 5th level slots into sorcery points then into 1st level slots for a maximum number of casts (+his 6th, 7th, 8th and 9th spell slots). If the sorcerer is level 20 (and has that boon that gets you one more high level slot), he could cast a total of 42 more spells, for a grand total of 95 possible surges in a single day (leaving 11 unused turns to account for surges like the one who makes you lose a turn in potted plant mode).

  2. By 14th level, a Wild Magic Sorcerer can roll twice on the Surge table and pick his preferred effect. Considering that there's 50 effects, the chance of being able to pick the height change surge with one of two rolls is 3.96%. In other words, it takes 25.25 surge rolls in average to get the height change surge (or 252.5 surges to have a +5 height increase... or 50.5 surges for +1 inch). Combining that with the info of point #1, we can almost get +2 inches every single day if I'm not mistaken.

  3. However, some other surges affect the calculation.

35-36 Roll a d10. Your age changes by a number of years equal to the roll. If the roll is odd, you get younger (minimum 1 year old). If the roll is even, you get older.

This one has the same average of +5 over 10 rolls as the other, but means that you will eventually die of old age by surging too much. Fortunately, the probability of getting this surge (rolling it on both results) is only 0.4%, so it takes 250 surges in average to get this effect once (or 2500 to have the average +5 age increase). There is, fortunately, another surge to reincarnate in a new (young) body, but that also resets your size to normal, so we don't want that either. So I suppose there's a soft limit on how much height you may gain with the technique, as you will eventually die of old age or have to reincarnate (unless you are immune to magical aging, but I don't remember how you can get that immunity).

01-02 Roll on this table at the start of each of your turns for the next minute, ignoring this result on subsequent rolls.

If this effect is gotten (every 25.25 surges in average), a lot more surges will happen in a short span of time, increasing the total amount of daily possible surges (and thus, the amount of daily possible height changes).

59-60 You regain your lowest-level expended spell slot.
99-00 You regain all expended sorcery points.

If one of these is gotten, that amounts to more available spell slots in the end, and therefore, more surges per day too.

The problem is I'm not good enough at math to make a good simulation, considering all above criteria, of an average Wild magic height change per day. Can you help me out ?

Additional clarifications :

  1. Only RAW is to be used (no unofficial way of increasing/decreasing the average height change).

  2. Assume peaceful but normal conditions (the Wild Mage is never interrupted in his 8h streak of rituals and casts, but in the other hand, he did not sell his soul to the devil nor receive a stressful wish to increase his chances of having the height change surge or anything similar).

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  • \$\begingroup\$ Is this asking for the probabilities of each outcome? the result of the most likely outcomes? or the most extreme outcome possible in a day? \$\endgroup\$ – GcL Aug 31 '18 at 20:29
  • \$\begingroup\$ The average daily height increase you can expect to get, considering the average possibility of getting the surges that affect how many spells & points you have (for more surges), and the avg possibility of getting the 10-surges-in-a-row surge. \$\endgroup\$ – Gael L Aug 31 '18 at 20:32
  • \$\begingroup\$ Also worth noting if you want to be super-precise... there's a small chance of being sheeped, which would make you unable to cast for the duration of a Polymorph spell. Fortunately, the reincarnation only kicks in if you die shortly thereafter. \$\endgroup\$ – Ben Barden Aug 31 '18 at 20:36
  • \$\begingroup\$ I had omitted the Polymorph effect, as you must save against your own DC, but if you wield a Staff of Power (+2 to saves), are Resilient (Wisdom) and don't have a huge Charisma, it's less likely to fail the save. Alternatively, you can always have a buddy close to you ready to counterspell the Polymorph. \$\endgroup\$ – Gael L Aug 31 '18 at 20:39
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    \$\begingroup\$ What happens if your height goes down to 0 inches, or below? And can you still cast normally when you're 1 inch tall? Or 1 year old, for that matter? (The reason I ask is that the game you're playing has a pretty high variance of results compared to expected gain, so the risk of gambler's ruin is also high.) \$\endgroup\$ – Ilmari Karonen Sep 1 '18 at 10:38
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First off, let me say that you don't have to use rituals to trigger Tides of Chaos, and it's going to be a lot slower, you can just do it by casting a 1st level spell, why Ritual cast it and waste time? You can't even use all of your ritual casts in an 8 hour time span that you get from being level 20.

Answering the question (WITH RITUAL CASTING): Ritual Cast takes 10 minutes + 1 Action.

Level 20, you get 20 Sorcery Points. You also get 85 Sorcery Points from higher level spells. 105 Sorcery Points total. 1/Long Rest: Regain all Sorcery Points. [pointless for Rituals when you don't get enough time to use it in a day]

Costs 2 Sorcery Points to gain a Comprehend Languages Spell. 4 natural casts, 52.5 casts from Sorcery Points

480 minutes in 8 hours 10.1 minutes to Ritual Cast a spell 47 Ritual Casts

You can cast Comprehend Languages 47 times in a day. That's 47 Surges. Note: you still get to roll a d20 on each spell you cast or 5% to get 2 Surges per cast; this equates to around 2.35 extra Surges each day. 49.35 Surges total per day.

(Average of the d10 being used is [-1,2,-3,4,-5,6,-7,8,-9,10]/10 = [5]/10 = 1/2.) Each Height roll grants you, on average, +1/2 inches to your Height. With 25.25 Surges equating to the roll you want, then you get, on average, +1 inch every 1.023 days.

More surges a day doesn't matter when you can't cast them since you're doing Rituals.

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