# Ernie and the Thrones of the Aqua-ant

Ernie can be uncomfortable at times in the company of other humans. He has never quite understood the minutiae of normal social interaction and often finds people to be puzzling and perplexing, their actions baffling and bewildering, and their motives obscure and obfuscated. So it is not surprising that he is drawn to animals for companionship.

I wasn’t surprised to see a large and apparently empty aquarium appear on his living-room table about two and a half months ago. “Lovely aquarium” I said when I saw it, “what are you going to put in it?”. “I’m not going to put anything in it because it is already inhabited” , he replied. “Just look more closely”. Despite profuse squinting with my nose right up to the glass all I could see were a few tiny specks moving languidly through the water. After a few minutes Ernie took pity on me. “They are a bit cryptic aren’t they?”, he said and set up his ‘self-focusing, auto-tracking, video-microscope’.

The enormously magnified dots revealed themselves to be some type of aquatic water-creatures and I spent quite a while entranced by the images displayed on the video screen. ”Sea-monkeys? Krill? Shrimps? Tiny crabs?”, I questioned. But I was wrong on all counts. “Actually” said Ernie proudly, “they are Malamerm Aqueus from Kzijekistan, although I prefer to call them aqua-ants. The tiny red one is the colony queen, and the blue one is the king. All the little green ones are minions. And this is the only colony outside of Kzijekistan”. I showed a dubious face and Ernie went on to explain.

“As you might know, along with Fleovium-298 the Kzijekistanians discovered a second isotope – Fl-303, which is much more radioactive. It has been utilized at the Kzijekistanian National Fission Reactor to provide electric power for the whole country. To minimize costs, they converted Lake Kyshtut into a cooling pond for the reactor. Unfortunately, cost-cutting and miss-management by the Committee for Bureaucracy and Administration resulted in serious radio-active discharges into the lake. By chance, during a recent clean-up operation, these tiny creatures – previously unknown to science – were discovered”.

Ernie went on to explain that the ‘aqua-ants’ were a bizarre mutated cross-breed between ants and water-snails. They were the only social, amphibian, insect/snail in existence.They had incorporated large amounts of non-radioactive FL-298 into their shells so were extraordinarily heavy for their size, and that their only known habitat was Lake Kyshtut. Due to Ernie’s renown as a Malacologist and Myrmecologist he had been sent a number of eggs in the hopes that a second breeding colony could be set up to protect the species from possible extinction. “And are they breeding yet?”, I asked. “No”, Ernie replied, with a worried sigh “seems that something isn’t quite right in their environment”.

A few weeks later I visited again. The aquarium was still on the table, but this time there was a shimmering translucent block of material floating on the surface of the water. Ernie explained. A local expert in Kzijekistan had discovered the secret to getting the aqua-ants breeding. The aqua-ants needed to participate in a curious mating ritual. Ernie explained: “Because they are so tiny, and live in the turbid water of Lake Kyshtut, the king and queen have difficulty finding each other, so the queen climbs onto any suitable piece of flotsam on the surface of the water. In a wonderful display of insect cooperation, minions attach themselves to various parts of the debris in order to tilt and rotate it in such a way as to lift the queen above the water’s surface so she can spy the king and signal him for a tryst. So I have put in this artificial spider-silk bubble that I had lying about. It is perfect cube, 80 mm on a side. If you watch for long enough you may be lucky enough to see the queen climb onto her throne”. And sure enough, after half an hours patient observation, I saw a tiny red spot attach itself to one part of the cube, following which a number of green spots attached to other parts of the cube, the cube slowly rotated and the queen was raised into the air. “And are they breeding yet?”, I asked again. “No”, Ernie replied, with a worried sigh “seems that something still isn’t quite right”.

Two weeks ago Ernie called me up in excitement. He had been invited to Kzijekistan to participate in the 1st Annual Conference on Malamerm Aqueus. He hoped to get in touch with the local experts to see if they could provide more advice to make Ernie’s colony a success. Would I mind ‘feeding the animals’ in his absence? Well, how could I refuse?

Just last night I was at Ernies, and had just finished with the feeding, when the phone rang. It was Ernie: “Thank goodness I got hold of you”, he blurted without even a hello, “there has been a disaster here in Kzijekistan. A fish truck plunged off the road while crossing the Kyshtut River and a number of red herrings were inadvertently released into the wild. They found their way to Lake Kyshtut and appear to have eaten every last aqua-ant in the Lake. It is feared that my colony is the only one left in existence.” I started to express my condolences, but Ernie interrupted. “But that isn’t the worst”, he continued, “the local experts believe that if a king and queen don’t meet within 90 days they will declare a vow of chastity – and will never breed. My colony is nearly that old already so there is no time to waste”. “Tell me how I can help”, I asked and listened for Ernie’s instructions.

So here is my problem:

The king, queen, and the numerous minions each weigh precisely 1.000 g. They are so small and dense that their buoyancy can be ignored.

The water in the aquarium has a density of precisely 1.000 g/ml.

The cube is precisely 80.000 mm on a side and is effectively weightless as will be the spherical bubble.

The king and queen will be raised by the minions (each on their own separate throne) so each is balanced at the absolute highest point possible above the water surface.

The minions can swim around in order to get the flotsam into position but once balanced they can only use their weight to hold it in position.

They have to stay in balance for up to an hour to ensure the royal couple can see each other, so the position has to be a stable equilibrium.

All I need to know is what diameter (to the nearest micron should be enough) I need to blow the bubble so the king and queen are both raised to exactly the same height on their respective thrones, and I would hate to be responsible for the species going extinct!

Can anyone help?

• A Mirabu asked (in an answer) In a valid "stable equilibrium", does the cube face on which the queen is positioned need to be oriented parallel to the water's surface? - to which I replied: No. the queen isn't necessarily on a face - she will cling to whatever part of the shape that can be raised highest. Similarly, the minions attach themselves at the optimal position/positions for raising the queen highest above the water. And the block will be at whatever possible orientation raises her highest - so long as it is a stable balance. – Penguino Jun 27 '16 at 2:51
• First of all, I can't get enough of these Ernie puzzles. I wish I could upvote them more. Second, I am thinking this is a problem involving the old metacenter or metacentre concept. Now to do some calculations. – axavio Jun 28 '16 at 2:13

If I am on the right track I can expand this answer....
I think that

The queen's throne will sit at 117.7632mm above the surface of the water, using a "corner down" orientation of the cube (she sits at the opposite high corner) and 5 minions holding on at the submerged corner. There are (might be?) multiple solutions for the diameter of the king's sphere, depending on how many minions want to join the party. I chose to go with the minimum number that would ensure positive stability, which is two minions holding on a the low point, and a spherical bubble diameter of 121.768mm, putting the king at near the same height above the water as the queen.

The queen:

There are 3 orientations of the cube that might work (a face parallel to the water surface, an edge submerged parallel to the waterline and a "corner down" orientation). At one gram per aqua-ant, we are not talking about huge volumes of displaced water, so it seems likely that the corner down approach (with a $80 \sqrt 3$mm distance from "keel" to highest point is the way to go.
The volume of water displace is $1ml = 1000mm^3$ per aqua-ant, queen plus minions. I used calculations adding one minion at a time, which gave the volume of water displaced, leading to the depth of the submerged corner based on volume of a pyramid (the submerged section). Assuming the minions were holding on at the lowest possible point gave the center of gravity of the system (queen at top, minions at bottom, cube weightless).

Now for the question of stability. A partially submerged object is stable if its metacenter is higher than its center of gravity. To calculate the metacentric height one needs to know the center of buoyancy (basically the centroid of the submerged volume, a pyramid in this case) and the second moment of area of the displaced waterline (a square in this case). I used the formulas from this page. It turns out that with up to four minions, the center of gravity is above the metacenter, i.e. an unstable situation. Beginning with 5 minions, though, the system is stable. Since the goal was to have the highest possible queen, I chose 5 minions to minimize the system weight and maximize her height. Result: top corner is 117.763mm above the water.

King:

The king is a different matter. We now know his height above the water, but need to calculate a bubble diameter that gives a stable solution with $n$ minions. The principle is the same as for the queen, but the depth to be submerged and the center of buoyancy for a spherical cap (the submerged volume) are different from the pyramid used on the cube. Its a bit more complicated since there is a additional degree of freedom. The closest I have come at the moment is a diameter of 121.768mm with 2 minions hanging on at the bottom of the sphere, which is submerged 4.0045mm, giving a king height of 117.7635mm (0.3 microns higher than the queen). This is a stable configuration with the CG some 20+ mm below the meta enter.

• You are on the right track for the queen, but I think you can do a leetle better with the king. – Penguino Jun 29 '16 at 21:26
• Remember, the minions will attempt to raise the king as far as possible (i.e. they won't attempt to match the queen's height). If you feel a more accurate diameter is necessary to meet your interpretation of the 'rules' that is OK. – Penguino Jun 29 '16 at 21:32
• Same numbers I came up with. Instead of calculating the meta-center I used a conservation of energy argument to determine how many minions were required to stabilize the royalty. – Penguino Jun 30 '16 at 8:20

It seems to me that in order to get stable equilibrium, the cube (plus queen and minions) needs to have its centre of mass below the surface. The way to get the queen as high as possible entails making a body diagonal vertical, and the queen being at its top. That body diagonal is $80\sqrt3$ mm long, and so the queen will be just shy of $40\sqrt3$ mm above the surface. Applying the same principles to the sphere, we need a sphere of radius just greater than the above figure, and thus of diameter double that, namely $80\sqrt3$ mm $\sim 138,564$ microns.

• You are working in the right direction but, at a diameter of exactly 138,564 microns for the spherical throne, I fear the aqua-ant royalty will choose the course of abstinence and the species will die out. – Penguino Jun 27 '16 at 21:05