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Problem 212


Problem 212


Combined Volume of Cuboids

An axis-aligned cuboid, specified by parameters { (x0,y0,z0), (dx,dy,dz) }, consists of all points (X,Y,Z) such that x0 ≤ X ≤ x0+dx, y0 ≤ Y ≤ y0+dy and z0 ≤ Z ≤ z0+dz. The volume of the cuboid is the product, dx × dy × dz. The combined volume of a collection of cuboids is the volume of their union and will be less than the sum of the individual volumes if any cuboids overlap.

Let C1,…,C50000 be a collection of 50000 axis-aligned cuboids such that Cn has parameters

x0 = S6n-5 modulo 10000
y0 = S6n-4 modulo 10000
z0 = S6n-3 modulo 10000
dx = 1 + (S6n-2 modulo 399)
dy = 1 + (S6n-1 modulo 399)
dz = 1 + (S6n modulo 399)

where S1,…,S300000 come from the “Lagged Fibonacci Generator”:

For 1 ≤ k ≤ 55, Sk = [100003 - 200003k + 300007k3]   (modulo 1000000)
For 56 ≤ k, Sk = [Sk-24 + Sk-55]   (modulo 1000000)

Thus, C1 has parameters {(7,53,183),(94,369,56)}, C2 has parameters {(2383,3563,5079),(42,212,344)}, and so on.

The combined volume of the first 100 cuboids, C1,…,C100, is 723581599.

What is the combined volume of all 50000 cuboids, C1,…,C50000 ?


立方体组合体积

一个与轴平行的立方体可以用参数{ (x0,y0,z0), (dx,dy,dz) }表示,包含所有满足x0 ≤ X ≤ x0+dx、y0 ≤ Y ≤ y0+dy和z0 ≤ Z ≤ z0+dz的点(X,Y,Z)。该立方体的体积为乘积dx × dy × dz。一系列立方体的组合体积是它们的并的体积;如果其中有立方体相交,这些立方体的组合体积将会比各自体积之和要来得小。

记C1、……、C50000是一系列共50000个与轴平行的立方体,其中Cn的参数为

x0 = S6n-5 modulo 10000
y0 = S6n-4 modulo 10000
z0 = S6n-3 modulo 10000
dx = 1 + (S6n-2 modulo 399)
dy = 1 + (S6n-1 modulo 399)
dz = 1 + (S6n modulo 399)

这里的S1、……、S300000来自如下的“延迟斐波那契生成器”:

对于1 ≤ k ≤ 55,Sk = [100003 - 200003k + 300007k3]   (modulo 1000000)
对于56 ≤ k,Sk = [Sk-24 + Sk-55]   (modulo 1000000)

因此,C1的参数为{(7,53,183),(94,369,56)},C2的参数为{(2383,3563,5079),(42,212,344)},依此类推。

前100个立方体C1、……、C100的组合体积是723581599。

全部50000个立方体C1、……、C50000的组合体积是多少?