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The short answer is: Because the ICE IS BLUE. Now we have to explain why it seem perfectly transparent on ice cubes and industrial ice blocks.

It have to do with the fact that most transparent materials are not perfectly transparent, and do absorb (and/or scatter) part of the light that goes trough them. And when the transparency is better for one specific color (wavelength), the material will look of that color, like a tinted glass that gives a tint to the white light that goes trough it.

Now the important fact is that "tint" effect is stronger the thicker the material is. Let's take a standard glass as a example: When looking trough the glass it seem perfectly transparent, because the thickness is too small to make its color noticeable. However, if you look at the edge of the glass, it looks green, something that show that glass is indeed green, but that became noticeable only when looking trough a section thick enough. enter image description here

In the same way, glacier ice is blue, but that can only be appreciated if the piece of ice you are looking at is thick enough. We can see this effect in the following picture of an iceberg, were sections of different thicknesses shows colors grading from transparent to blue. enter image description here

The remaining questions is: Why they are not always blue? As we know many icebergs are white. The same is trough for snow, that is ice after all.

The key here are the bubble. And the reason is that on each ice/air interface, part of the light gets reflected and only a fraction goes trough. Therefore, the most bubbles a piece of ice have, the less the light will be able to travel trough it before bouncing back out. And the less ice that light have gone trough, the less the tint received. I've made this figure to help understanding the idea: enter image description here

On the left, a piece of ice with very few bubbles allows a longer travel path trough the ice, inducing a marked blue tint in the outgoing light. On the right, in contrast, if there are many bubbles the light will go back right away with a very subtle or no tint at all.

Snow id the extreme of it, witch it is mostly air (bubbles), and some small pieces of ice, that's why it looks so white.

An interesting ramification of this, is that those deep blue pieces of ice have started as snow. But as snow piles up, the weight of the snow on top compacts it, squeezing the air out until you end with a block of ice with bubbles (which is officially the point it start to be considered ice, before it is snow or firn). In a big glacier, hundreds or thousands of meters of ice can exert such a pressure that the bubbles get compacted becoming smaller and smaller until they disappear and the air gets dissolved in the ice. Therefore, the more bubble-less and blue a piece of ice is, it means it comes from deeper in the glacier, meaning also that comes from further up in the glacier basin and it is, therefore, older. So, now you can identify deep, old ice just by its color.

To give a more familiar example of how adding bubbles can turn something transparent with a tint into something white, take the case of egg white, which is trasparent with a yellow tint. But as why whip it, adding bubbles, it to turn it into meringue that is perfectly white. enter image description here

However, there is a trick to make the color of the seemingly-white snow noticeable. And that is letting a ray of light to bounce many times on it, each time it will travel through a little bit of ice getting a bit of blue tint, and after enough times it will look unquestionably blue. That's what you see when looking at the entrance of a snow cave, or a crevasse in a glacier. The light that comes from deeper in the cave have bounced more times in the walls becoming bluer: enter image description here