3 ms·

There were 4,096 total layers; currently 262 layers have been removed with 17,758,801 cubelets left on the current layer. So 3,834 layers and 112,689,867,042 c

There were 4,096 total layers; currently 262 layers have been removed with 17,758,801 cubelets left on the current layer. So 3,834 layers and 112,689,867,042 cubelets remain. The cube age is 166 days, so at the current rate of progress (~148,787,753 cubelets per day), it should take another 758 days to complete. However, the rate has been slowing from over 2 layers/day at the start to just over 1.5 layers/day. Combined with user attrition and this new ability to add/remove layers, it's hard to say when exactly it'll be completed.

danbruc 14y ago
There are only 2048 layers. I used the statistics from [1] at 5:20 from day one. Layer 1, 66,631,501 cubelets remaining on the current layer and 33,982,651 cubelets destroyed in total. Because it is the first layer it is made up out of 66,631,501 + 66,631,501 = 100,614,152 cubelets. An estimate of the edge length is obtained by dividing by 6 faces and taking the square root and this yields 4,095.0000407. It is a bit counterintuitive - at least for me - but the actual edge length is 4,096. I would have expected that counting the edges twice and the corners trice becomes insignificant for large edge lengths but if you do the math you will see that this approximation has the limit n - 1 as n goes to infinity. So because every layer adds 2 to the edge length there are 2048 layers. That is bad - there is no central cubelet because the edge length is not odd. Maybe this is the hidden secret. I guess they used 4096 because it simplifies texturing the thing, implementing level of detail or stuff like that, does it? Or they have 8 secrets in there to make it a tiny bit more fair. One more thing, the number of cubelets on the surface of a cube with edge length n is n^3 - (n - 2)^3 - the whole cube minus the smaller cube under the surface. Plugging in 4096 yields the number obtained from the statistics so we probably made no mistake. Okay, let's fix the other numbers, too. The cube started with 68,719,476,736 cubelets. I assume the current layer was 263 - the statistics gives the number of layers removed and layer 263 seems to be more consistent with the time of the comment - and there are 23,220,183,736 cubelets in those layers. Subtracting the 17,758,801 cubelets remaining yields 23,202,424,935 cubelets destroyed in 166 days or roughly 139.8 million cubelets per day. There are 45,517,051,801 cubelets remaining and assuming the same rate of destruction this yields roughly 326 days to go. [1] https://www.youtube.com/watch?v=giEykqDoUCQ https://www.youtube.com/watch?v=giEykqDoUCQ
- Dylan16807 14y ago
  I can help your intuition with the n - 1, I hope. It's true that it's negligible when you're handling cube roots. But when you're cutting down to faces you end up ignoring the entire inside of the cube, so edges can cause problems. Looking at a single face, you have four edges that must be shared, so give away two edges a keep two edges, and you very easily result in a square of n-1 on every face.
  - danbruc 14y ago
    I like the idea of embedding n - 1 squares in each face, it is quite intuitive...but not fully satisfying. It would be nice if exactly six n - 1 squares would fit into the surface...but no, there are two additional uncovered cubelets. Two is again unintuitive to me - 6, 8 or 12 and I would be happy. On the other hand unintuitve results make me redo the work to find the mistake and so unintuitve results are less likely wrong and that in the end is a good thing. After spending some more time thinking about my initial reasoning it became a bit more intuitive. I thought there are O(n^2) cubelets on each face and the error with the edges is only O(n) so the error should become negligible and it does - the linear error becomes a constant error of less than 1. I just expected it would approach 0 and this is not true. And now I will stop bothering - my intuition is not the best in this area and I will just have to calculate everything. I just realize that I can not even easily say if the approximation over- or underestimates...we use some cubelets twice, so we under...over...underestimate the area of a face...no, over?...under!...
- itafroma 14y ago
  You're right, it's 2,048 layers. The sad part for me is that I knew that number, but when I did the math, I couldn't get it to work with the current layer's numbers, because I forgot that the cube length goes down by two, not one. Oh well, shows I shouldn't do math late at night!

danbruc 14y ago

There are only 2048 layers. I used the statistics from [1] at 5:20 from day one. Layer 1, 66,631,501 cubelets remaining on the current layer and 33,982,651 cubelets destroyed in total. Because it is the first layer it is made up out of 66,631,501 + 66,631,501 = 100,614,152 cubelets. An estimate of the edge length is obtained by dividing by 6 faces and taking the square root and this yields 4,095.0000407. It is a bit counterintuitive - at least for me - but the actual edge length is 4,096. I would have expected that counting the edges twice and the corners trice becomes insignificant for large edge lengths but if you do the math you will see that this approximation has the limit n - 1 as n goes to infinity. So because every layer adds 2 to the edge length there are 2048 layers. That is bad - there is no central cubelet because the edge length is not odd. Maybe this is the hidden secret. I guess they used 4096 because it simplifies texturing the thing, implementing level of detail or stuff like that, does it? Or they have 8 secrets in there to make it a tiny bit more fair. One more thing, the number of cubelets on the surface of a cube with edge length n is n^3 - (n - 2)^3 - the whole cube minus the smaller cube under the surface. Plugging in 4096 yields the number obtained from the statistics so we probably made no mistake. Okay, let's fix the other numbers, too. The cube started with 68,719,476,736 cubelets. I assume the current layer was 263 - the statistics gives the number of layers removed and layer 263 seems to be more consistent with the time of the comment - and there are 23,220,183,736 cubelets in those layers. Subtracting the 17,758,801 cubelets remaining yields 23,202,424,935 cubelets destroyed in 166 days or roughly 139.8 million cubelets per day. There are 45,517,051,801 cubelets remaining and assuming the same rate of destruction this yields roughly 326 days to go. [1] https://www.youtube.com/watch?v=giEykqDoUCQ https://www.youtube.com/watch?v=giEykqDoUCQ

Dylan16807 14y ago
I can help your intuition with the n - 1, I hope. It's true that it's negligible when you're handling cube roots. But when you're cutting down to faces you end up ignoring the entire inside of the cube, so edges can cause problems. Looking at a single face, you have four edges that must be shared, so give away two edges a keep two edges, and you very easily result in a square of n-1 on every face.
- danbruc 14y ago
  I like the idea of embedding n - 1 squares in each face, it is quite intuitive...but not fully satisfying. It would be nice if exactly six n - 1 squares would fit into the surface...but no, there are two additional uncovered cubelets. Two is again unintuitive to me - 6, 8 or 12 and I would be happy. On the other hand unintuitve results make me redo the work to find the mistake and so unintuitve results are less likely wrong and that in the end is a good thing. After spending some more time thinking about my initial reasoning it became a bit more intuitive. I thought there are O(n^2) cubelets on each face and the error with the edges is only O(n) so the error should become negligible and it does - the linear error becomes a constant error of less than 1. I just expected it would approach 0 and this is not true. And now I will stop bothering - my intuition is not the best in this area and I will just have to calculate everything. I just realize that I can not even easily say if the approximation over- or underestimates...we use some cubelets twice, so we under...over...underestimate the area of a face...no, over?...under!...
itafroma 14y ago
You're right, it's 2,048 layers. The sad part for me is that I knew that number, but when I did the math, I couldn't get it to work with the current layer's numbers, because I forgot that the cube length goes down by two, not one. Oh well, shows I shouldn't do math late at night!