TAOCP 7.2.2.1 Exercise 351

Let $s = a+b+c+d+e$.

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Section 7.2.2.1: Dancing Links

Exercise 351. [M46] [M46] Can $5^5$ hypercuboids of size $a \times b \times c \times d \times e$ always be packed into a 5-dimensional hypercube of size $(a + b + c + d + e) \times \cdots \times (a + b + c + d + e)$?

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Let $s = a+b+c+d+e$. We are asked whether it is always possible to pack $5^5$ copies of the 5-dimensional hypercuboid

$$ a \times b \times c \times d \times e $$

into the 5-cube $[0,s]^5$.

1. Fundamental necessary condition: volume

Any packing of disjoint axis-aligned hypercuboids that exactly fills a hypercube must preserve volume. Hence a necessary condition is

$$ s^5 = 5^5 , a b c d e. $$

But in general,

$$ (a+b+c+d+e)^5 \neq 5^5 a b c d e. $$

Therefore the claimed packing cannot exist for arbitrary positive $a,b,c,d,e$.

This already disproves the statement “always”.

2. Explicit counterexample

Take

$$ (a,b,c,d,e) = (1,1,1,1,2). $$

Then

$$ s = 6,\quad s^5 = 6^5 = 7776, $$

while the total volume of $5^5 = 3125$ hypercuboids is

$$ 3125 \cdot (1\cdot 1\cdot 1\cdot 1\cdot 2) = 6250. $$

Since

$$ 7776 \neq 6250, $$

a perfect packing into $[0,6]^5$ is impossible.

Thus the statement fails.

3. Why the inductive construction in the proposed solution cannot be repaired

The flawed proof attempted to partition the cube into 5 slabs and apply a 4-dimensional induction hypothesis. Even if the arithmetic error is fixed, a deeper issue remains:

  • Each slab $S_x$ has geometry $I_x \times [0,s]^4$, where the 4-dimensional side length is $s$.
  • Any valid reduction would require packing into a 4-cube whose side length matches the sum of the remaining edge lengths.
  • However, no consistent decomposition aligns the volumes or shapes so that each slab contains a correct scaled instance of the same problem.

Thus the inductive structure does not correspond to the actual geometry of the target packing problem.

4. Conclusion

The statement is false.

$$ \boxed{\text{It is not always possible to pack } 5^5 \text{ such hypercuboids into } [0,a+b+c+d+e]^5.} $$

A necessary volume condition fails in general, and explicit counterexamples exist.