Time to draw some diagrams. Define a sphere of diameter D, at a distance d from a point U. In this scenario, D is a star's Photosphere, and U is your eye. Assume that all light given off by the disc is radial (meaning on a straight line between the center of the sphere and each point on the surface of the sphere). With this model, how many lines intersect at U?
One.
If that point is giving off 1,000,000 photons steadily every second, point U will receive 1,000,000 photons steadily every second. Right? With me so far? Ok, given this simplified model, how does distance, d, reduce the number of photons reaching point U? (hint: think of the words:"it don't"). Increase d by a factor of a million, reduce it by a factor of 1000, it doesn't matter, there are still the exact same number of photons going through point U every second. Olber's Paradox is (in this simplified form) the question of why, if the number of stars is infinite, why doesn't every line of sight from point U end up on a surface of one of those infinite number of spheres? Working with this simple model, the ONLY solution in a static Universe (no dust, no inflation, etc.) is that another sphere between U and the first sphere partially eclipses that first sphere so that the photon from the first sphere hits the closer sphere. You can see, perhaps, how each sphere only contributes one single line which will intersect at U, but will obscure many lines heading towards U from farther away. Is this the solution to the paradox? No, unfortunately it isn't. We need to go back to the assumption that only radial lines emanate from each sphere. In reality, each point of a star's Photosphere gives out a lot of photons in all directions (not just radially). So, we need a new diagram. This one with spheres of radius D and various distances from point U (we still assume your eye is a "point-sized" detector, we could treat it otherwise, but this is good enough if we are careful.) So, instead of just lines, this diagram needs areas and lines. Draw as many spheres as you want, at various distances and with varying "overlap"(not physically on top of each other but in each other's line of sight). Now, you need to draw two lines for each sphere, the lines are to be tangent to the surface (tangent to the circle) AND intersecting point U. For each star, you can only draw (on paper) two such tangent lines, one on either edge. Draw them one at a time. Start with the one nearest U.
With your straightedge, line up one edge of the circle (tangent) and point U. Starting at point U draw a line segement which goes to the circle and continues on to the edge of the paper (thorugh any stars). Repeat with the other edge of that circle. These two lines represent two things: first all of the light from the circle to U and second, the shadow of that Star (for areas behind it will be in that star's shadow and not visible to U). Shade in the area from that circle to U outlined by the two (intersecting at U) lines and the circle of the star. This represents the photons going from the surface to point U. Using a different color, shade in the area in back of the circle (the 'shadow'). From this area no light can get past the star and hit U, so no lines need be drawn in this area. Repeat the same procedure with the next closest circle. There are now four possibilities: 1. The star is clear of any closer star and so you do the same thing as before or 2. The star is partially in the shadow of a nearer star, with one edge exposed. In this case the area to be shaded is defined one tangent line and the edge of the shadow you've already shaded. As before shade in the light area, and the shadow area in back of it, using the same two colors (or cross-hatches, or patterns, etc.) 3. The third possibility is that the circle is partially
obscured by two tangent lines, and if so you color in the light area and also the shadow in back using the shadow lines as a guide. 4. The star is completely in one circle's shadow. Do nothing in this case.
Continue until done with all of them. Perhaps you can see that if you drew enough circles and placed them randomly (here we assume they're all the same diameter, D) that eventually the point U would be COMPLETELY surrounded by light. IF you also assume (correctly) that each point from every star (every circle) is giving off the same number of photons, then just like in the first drawing, the number of photons from each "line of sight" is equal REGARDLESS of distance. If one star isn't sending light toward U, then a star in farther away (or closer) is, for ANY line you care to draw between U and infinity. Distance is not the problem. If we include dust and other things which create shadows between us and the stars, then of course this analysis is wrong. But the dust isn't (we think) the major issue with Olbers Paradox. Again, to be clear, the inverse square law has no (significant) part in the solution: distance isn't an important part (in a static non-expanding Universe). Boy, I've set up the problem, but have run out of patience and won't write the answer. Others have already, but there seems to be much that is misleading, also.
Hints to solve "paradox".
1. The first stars, we think, are only ~13½ billion years old (the Universe is NOT static), stars farther away (and we do think there are stars farther) than 13½ billion light years can NOT contribute (yet) to the light we see.
2. The Universe is expanding, meaning that the distances between galaxies is increasing, meaning there ARE some lines of sight which will never intersect a star's surface (in a finite amount of time). Another way to put that is that many lines of sight between us and the 'edge' of our Observable Universe do NOT intersect a star.
3. Most of the Milky Way Galaxy can NOT be seen from Earth - there IS too much dust in the way (at visible wavelengths).
4. Because the Universe is expanding, light does what we call "redshift" (as it travels through billions of light-years, near by (within thousands or hundreds of thousands of light years) light is affected so little, that you wouldn't notice. Redshift means blue light turns red, red turns into radio waves... far enough (long enough) and almost ALL the light will be converted into microwaves... There IS no Olber's Paradox. If we could see microwaves, then we would see the sky lit up like daylight all the time. (This is called the Cosmic Microwave Background).
