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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.

[![enter image description here][1]][1]

“As you might know, along with [Fleovium-298][2] 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.

“It is all very simple” he replied. “Research completed just before the disaster now shows that the king climbs on a second piece of floating debris and is also lifted into the air by the minions. Once they can see each other they will dive back into the water and breed”.  “So you want me to put another cube into the water?”, I asked. “Not possible”, Ernie replied, “I only made one before the cubical-bubble gun went on the blink, so you will have to make a spherical bubble.” I felt confident about this – I had used Ernie’s [bubble][3] gun in the past and since he had optimized it’s design it was a simple process of dialing up the required diameter of the bubble and pressing the green button. I was just about to go when Ernie added one last comment. “Make sure you choose the diameter so the king will be raised to the exact same height as the queen. If the king is higher (even by a micron), the queen will assume he is misogynistic. If the king is lower, she will assume he isn’t genetically worthy. Either way she will refuse to breed and the colony will turn celibate. So the diameter of the bubble should be precisely…” – and at that point the phone went dead. I waited but Ernie didn’t call back. And when I checked on the web I discovered that ‘*thunderstorms over Kzijekistan are likely to disrupt communications and transport for the next few days*’.

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?

[1]: https://i.sstatic.net/lneIm.jpg
  [2]: https://puzzling.stackexchange.com/questions/8763/ernie-and-the-island-of-stability
  [3]: https://puzzling.stackexchange.com/questions/17900/ernie-and-the-bubbles-of-lucretia