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In 3.5, black puddings possess:

Split (Ex) Slashing and piercing weapons deal no damage to a black pudding. Instead the creature splits into two identical puddings, each with half of the original’s current hit points (round down). A pudding with 10 hit points or less cannot be further split and dies if reduced to 0 hit points.

Does this "half" indicate a current or maximum hit point total?

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    \$\begingroup\$ Haha! I saw this on the list of popular network questions and clicked it to see if it needs to be migrated to Cooking :P \$\endgroup\$ – rumtscho Jun 15 '14 at 17:27
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    \$\begingroup\$ I clicked it because I wanted to find out whether it was possible to just sort of glom the two back together. \$\endgroup\$ – octern Jun 15 '14 at 22:04
  • \$\begingroup\$ Added the [pathfinder] tag since the rules are literally word-for-word identical in both systems and it had been asked for Pathfinder. \$\endgroup\$ – KRyan Mar 13 '17 at 2:13
  • \$\begingroup\$ This is how you get chocolate pudding in cups; just like starter sourdough, you keep 1 pudding alive in storage and cut off pieces to put in the cups for sale. Cruel, really. \$\endgroup\$ – CaM May 31 '17 at 20:19
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The split puddings’ current HP is no doubt half the original pudding’s current HP prior to splitting:

each with half of the original’s current hit points

But what does this mean for its max HP?

The statement only covers current HP; max HP is not mentioned. Since the creature “splits into two identical [black] puddings,” with the only modification given being that its current HP is half the original’s current HP, I would otherwise use the black pudding creature entry to determine the remaining statistics. That is, it has a regular black pudding’s max HP, and yes, can be healed back up to it.

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    \$\begingroup\$ So a cleric with a pointy stick can raise a black pudding army in minutes \$\endgroup\$ – G0BLiN Jun 15 '14 at 15:42
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    \$\begingroup\$ @G0BLiN If he can control them (there is the Ooze Domain for that, but it works like Rebuke Undead, i.e. it has limits), and he doesn't mind spending a lot of spells on healing, sure. Spawning undead seem like a better choice here; you can use them as lieutenants so you don't have to directly control the spawn, so you can get far more, and undead tend to be a lot more dangerous than black puddings. \$\endgroup\$ – KRyan Jun 15 '14 at 15:44
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    \$\begingroup\$ It does suggest a very cruel trap ("There seems to be a conjuration/healing effect on the floor, and an egg-slicer-like apparatus near the ceiling - I wonder what's supposed to come out of the grate above..." - famous last words) \$\endgroup\$ – G0BLiN Jun 15 '14 at 15:50
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    \$\begingroup\$ A bit tangentially, the NetHack computer game, which draws heavily from early D&D editions, implements black pudding splitting exactly like this. And, yes, the consequences are predictable. \$\endgroup\$ – Ilmari Karonen Jun 15 '14 at 16:16
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    \$\begingroup\$ @IlmariKaronen Oh trust me, NetHack was strongly on my mind when I wrote this answer. And "the DevTeam has arranged an automatic and savage punishment for pudding farming. It's called pudding farming," was basically my first reaction to G0BLiN's comment. \$\endgroup\$ – KRyan Jun 15 '14 at 17:10
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I would suggest that a second, equally valid interpretation to the splitting of 'black pudding' is that the actions of slashing and piercing weapons works to divide the matter into two halves of the original. That is, that the 'half hp of original creature' becomes the max hp for each halve.

This redistribution of matter, where 10hp is the minimum hp required to form a coherent whole, is different from the reduction in hp until it hits this minimum state of maximum integrity.

In descriptive terms, once a part is reduced below 10hp, parts of the matter become inert. Hence, healing the creature would likely only serve to bring it back to its maximum segment hp - that of 10hp at the lowest.

A little off topic but I would also suggest that a healing could also be used to bring two coherent wholes into a single whole of double the hp, presuming that the original creature could in fact do so - otherwise it would take a greater healing spell capable of healing more permanent injuries (or regeneration) to do it.

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