Prof. Will Lestrange

The glass is the right size, but the water has evaporated from all the spellcasting! You summon a glass of water from a different room with Accio.

The Demiguise cautiously sniffed the fruit and took a step closer towards me.

Ebony

Norway maple

I tried talking to the Demiguise but it didn't understand my language.

Wingardium Leviosa + Protego - Aguamenti: 17 Protego: 16 Reparo: 13 Wingardium Leviosa: 14

An unmarked vial near the master's station contains a clear, thick substance that does not look particularly impressive. But this substance, a special version of what Muggles call "glue" (the master we're visiting prefers to call it "Bonding Substance"), is an integral part of magical wands! A single drop of the substance, when placed on a feather, hair, or heartstring to be used as a magical core, will enable the core to have a special bond with the wand wood. The core can then be placed inside the wand wood to enable its magic to pervade throughout the entire wand! But if too much is added, the core will bond with itself instead of with the wand wood, causing spells cast with the wand to malfunction...

The Finite Incanatatem works fine, but the water is still in danger of spilling. You cast Engorgio to make the glass slightly larger so the water wouldn't spill!

I never really worried too much about refurbishing the shop before simply because people tend to get in and out of the shop so quickly that comfort seems to be less of an issue. But based on the description in the prompt, I think that some sort of cleaning and air-purifying charm, going on constantly, would keep the shop in a reasonable state. It would be much easier for Ollivander to make sure the charm was periodically maintained than to actually try and stop what he was doing to get actual cleaning done! And maybe another charm or two for ambiance might help too... playing some music, perhaps?

I would have thought the second possibility made more sense: while wands have magic of their own, I do not believe they can really act on their own free will as they require the magic of others to work properly. (If they did have their own free will, they would be more likely to react in some way when a Muggle tried to use them instead of just acting as motionless sticks, which is how Muggles actually see the wands). So it has to be some sort of magical compatibility in the way the magic of the wand matches up to the magic of the wizard - and that is only activated when the wizard actually does something to the wand. Notice that Harry Potter had to actually wave the wands himself to get any sort of reaction with them; if the choice was made by the wand, the wand could have likely been able to react - or not - independent of Harry's motion!

Reparo + Aguamenti - Aguamenti: 17 Protego: 18 Reparo: 11 Wingardium Leviosa: 14

My first thought came to the Imperius Curse, of course, but I cannot imagine it working on a phoenix: its healing and rebirthing powers have got to be able to work around wizarding spells - even those with fundamental magic! We also know that Fawkes has only given out two feathers throughout his entire life as a companion of Albus Dumbledore, and both were made into wand cores. From this, I have to assume that phoenixes do not readily part with feathers at all; they only would shed a feather when the time was right (and if the phoenix was in the presence of a wizard they trusted, the wizard would be able to hold onto and possess the feather; otherwise the phoenix may well just immediately incinerate the feather upon it being shed). That would explain why phoenix feather wands are so rare: if phoenixes so rarely shed feathers, there wouldn't be enough for a lot of wand cores!

But before long I realized I had no idea which direction I was going!

My tree is called the Tree of Five. It is known for its fractal shape: out of the tree trunk grows exactly five branches - and out of each of those branches there are five more branches, and so on and so on up to infinity. (Naturally, infinite numbers of arbitrarily small branches can only exist due to magic). The branches themselves have various leaves growing out of them; the leaves appear to change colour when you look at them which means that no one knows their true appearance or shape. While the Tree of Five has no fruit, the bark growing from the trunk can be boiled down to a delicious treacle!

A tree with strong ties to African folklore is the baobab tree. This tree, which is native to the African continent as well as other places scattered around the Southern Hemisphere, is known for its appearance; with its branches looking like roots, the plant appears that it was planted upside down! And there are various myths that attempt to explain why the baobab was planted that way: either as a punishment of some sort or because the animal tasked to plant it did not like its appearance. But despite its upside-down appearance, it's a much-revered plant for its fruits, leaves, flowers, bark, and seeds, many of which have various sorts of medicinal properties! And these days, the 'upside-down' form of the baobab only serves to make it more recognisable...

American holly

Sugar palm

It seems like the simplest thing in the world. You start with a positive integer; if it's odd, you multiply it by three and add one to get an even number. And if it's even, you simply divide it by two. Repeat this as many times as needed until you end up with the number one. If you were to continue from there, you'd end up repeating the sequence "1, 4, 2" forever. Easy, right? Let's start with the number 99. It's odd, so multiply by three and add one to get 298. Divide by two for 149, which is odd - so the odd rule will give you 448. From here, you divide by two until you can't do so anymore: 448 becomes 224 becomes 112 becomes 56 becomes 28 becomes 14 becomes 7. From here, you go to 22, which takes you to 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and finally 1 in sequence. It may have taken a while, but we ended up there in the end! And you can try the same thing yourself; no matter what positive integer you choose, I fully expect that you will end up at 1 yourself! It may take a while if you choose certain numbers, but any reasonable choice will get you there. Muggle Arithmancers aren't satisfied, yet, though. Ever since a Muggle Arithmancer named Lothar Collatz investigated this game nearly a century ago, they have wondered if there was any positive integer where you could play this game forever, but never make it to the number 1. (It is called the "Collatz conjecture" in his honour). Although they have determined that you can make it to 1 starting at any number up to two hundred ninety five quintillion or so, they still don't know if there's a number that's larger that lets you play the game forever without ever reaching the 1! That said, nearly all proper Wizarding Arithmancers who have played the game have been bewitched by the way the numbers in the sequence rise and fall like hovering crystal balls (Muggles would call them "hailstones"), guided by some mystical power we do not fully understand!

The phoenix feather flies up to the ceiling, along with the desk that it rests on! I cast a Cushioning Charm on the desk so that it doesn't crash-land onto the floor.

Since we don't know your real name, I think a good song to sing would be the "Banana Fana" name song! It is a four-line rhyming song that changes based on the person's name; the lyrics are as follows: <name>, <name>, bo <bame> banana fana fo <fame> fee fi mo <mame> <name>! where <name> is the name of the person you're singing about, and <bame>, <fame>, and <mame> would be that name with the first letter replaced by a B, F, and an M respectively. For example, if you were to sing the song for me, <name> would be "Will", but <bame>, <fame>, and <mame> would be "Bill", "Fill", and "Mill" respectively. This is a great song that you can adapt to each person entering into the common room - and it is personalised enough to make it apparent that you care for everyone who is passing by!

at least I beat an hour!