Why is ice made from boiled water clear?

The short answer: Cloudy ice is caused by gases (mainly nitrogen and oxygen) dissolved in the water that come out of solution when the water freezes. The small bubbles trapped in the ice cause the white appearance. Boiling the water removes the air dissolved in it, producing clear ice as a result.

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This answer was meant as a comment to @WetSavannahanimal aka Rad Vance but it is rather long and I hit the character limit. The reason for the opaque center should be due to the manner in which the water volume is freezing. Presumably the solution is not mixed and the outside freezes first forming a crystalline (ice) wall through which the gas cannot escape. As the wall thickens gas is released from the water that solidifies into the central solution that remains. This concentrates the gas in the remaining liquid in the centre. When the gas concentration in this solution hits the saturation value for the liquid at it's current state some of it exits the solution forming the cavities, simultaneously some ice should form, returning the solution to the saturation concentration. This is repeated until all the water is frozen. The observation that clear ice is made buy bubbling gas through it as it freezes, indicates that mixing of the solution allows the saturated gasses to escape from the surface of the total water volume as the solid forms rather then forming in the center. Now one might ask the question why is it that there isn't just a single bubble. The first reason, from a bulk solution point of view, is that the water is freezing incrementally forming bubbles as it goes. It is really oscillating about the equilibrium state of the solution, that is the gas saturation point of the freezing solution. The exact conditions of this point will vary slightly as the liquid freezes. The pressure that the ice in the center froze at is likely larger then the pressure at which the surface ice froze at for instance, similarly there is a the temperature at which it froze might also vary. There is probably also a concentration effect, that is as this equilibrium point shifts about the gas saturation point will shift about, this change in concentration also affects the freezing point a little. There are about four effects (Temperature, Pressure, Volume and Gas Concentration) at play during the freezing. The second effect, from a finite volume point of view, is that locally about the cavity the water might experience a "rush of gas" which could locally freeze a film of water encapsulating the bubble hence the final complex arrangement of cavities and not the formation of a single bubble. Now it might just be possible to see these freezing point differences using the polarizer trick. I've only seen this with clear plastics till now but it should work here aswell. Next time you go to a movie get a pair of 3d goggles. take out the two polarizing lenses and hold them on either side of the ice cube by rotating them a little you should see the internal crystalline structure of the ice as a mess of swirly lines. You should probably see more of a swirl or a scattering in the center near the bubbles indicating the localized changes in crystal structure. You should compare this to the transparent cube. There is a further trick you can try. If you controlled the freezing of the ice by some degree you could control the formation of the bubbles. For instance our ice trays are plastic and we tend to have a clear upper layer with bubbles forming in the lower part of the cube. I suspect the plastic is retaining it's heat and delays the freezing of the bottom and the side of the cube. I suspect if one warmed or even cooled the ice tray before forming the ice one might see a different formation of cavities (As shown in the second image here for example). If you used a metal ice tray you should see the effect you have observed. If you combined a metal and a plastic ice tray together you might get more bubbles closer to the plastic side. You might even be able to get a polka dot/explosion/blobbing effect by dotting glue inside a metal tray, or holding a metal rod within the volume as you froze it. Although you could just be adding points for the gas to form and escape resulting in clear cubes. Alternatively you might just get a carrat (hat). This Fellow seems to have done a lot of the leg work for you. Beneficially he seems to enjoy drinking his experiments afterwards (YMMV).

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If you are really interested you should see if NASA did any freezing experiments in space it might show other methods by which one could control bubble formation. Similarly you might find high speed camera footage of freezing during which you should see some interesting effects as the bubbles form (Though it's a bit of a contradictory use of these cameras and I doubt the people who own them have thought to use them for this purpose).

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