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 Pack the Box (Posted on 2004-01-04)
You have an empty box which is a 11cm cube. You also have a supply of blocks. You have 54 2cm cubes, 24 3cm cubes and two 5cm cubes. Pack all 80 cubes into the box.

 See The Solution Submitted by Brian Smith Rating: 3.3333 (3 votes)

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 solution | Comment 5 of 6 |
First, if the two 5x5x5 cubes were other than in two diagonally opposite corners of the cube, it would require more single-unit cubes than the one allowed by the 1080 cc taken up by the 2- and 3-cm cubes after the 250 cc of the two 5 cm cubes. For example, even if the two 5-cm cubes were touching vertex-to-vertex, three whole areas of 1-cm thickness would be created with area 5x5 along the faces of the non-outside 5x5x5 cube that don't touch the vertex that touches the other cube.

Working at it by trial and error, where 11 units can be 3, 3, 3 and 2, or 2, 2, 2 and 5, or 2, 2, 2, 2 and 3, placing cube by cube into the large cube (after the two 5-cm cubes were placed as mentioned above), I came up with a solution that is best described, after the fact, as follows:

A 6x6x5 rectangular solid subset can be formed from a layer of 4 3-cm cubes laid out in a square formation, overlaid by a layer of 9 2-cm cubes also laid out in a square formation. Six of these solids would then make the 6 x 9 = 54 2-cm cubes and 6 x 4 = 24 3-cm cubes.

So place one 4-cm cube and three of the 6x6x5 solids at the bottom of the 11-cm cubical box. One of the 6x6x5 solids will be lying flat (i.e., with its 5-cm dimension upward from the base) taking up a 6 cm by 6 cm square on the base. The other two are laid more vertically, with one of their 6-cm dimensions each vertically, so as to take up a 6 cm x 5 cm rectangle on the base. This occupies the full base partially to a height of 6 cm and the rest to a height of 5 cm. An identical, except mirror-imaged, complex of these 4 items can then be inverted and placed over the original set. This leaves an empty 1-cm cube in the middle.

When originally fitting these together, I did not realize the possibility of the fitting of 4 3-cm cubes and 9 2-cm cubes into a 6x6x5 package, but just tried fitting individual pieces. What resulted, however, was in fact describable in that modular form. The diagrams below show layer-by-layer (1 cm high) the numbered cubes which occupy each 1-cm unit, from top to bottom. The numbers are the order in which I placed the cubes. Each cubic centimeter is labeled with the cube number to which it belongs:
```
+--+--+--+--+--+--+--+--+--+--+--+

| 1  1  1  1  1|80 80|79 79|78 78|

+              +     +     +     +

| 1  1  1  1  1|80 80|79 79|78 78|

+              +--+--+--+--+--+--+

| 1  1  1  1  1|75 75|76 76|77 77|

+              +     +     +     +

| 1  1  1  1  1|75 75|76 76|77 77|

+              +--+--+--+--+--+--+

| 1  1  1  1  1|74 74|73 73|72 72|

+--+--+--+--+--+     +     +     +

|47 47|62 62 62|74 74|73 73|72 72|

+     +        +--+--+--+--+--+--+

|47 47|62 62 62|67 67 67|66 66 66|

+--+--+        +        +        +

|48 48|62 62 62|67 67 67|66 66 66|

+     +--+--+--+        +        +

|48 48|61 61 61|67 67 67|66 66 66|

+--+--+        +--+--+--+--+--+--+

|49 49|61 61 61|65 65|64 64|63 63|

+     +        +     +     +     +

|49 49|61 61 61|65 65|64 64|63 63|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

| 1  1  1  1  1|80 80|79 79|78 78|

+              +     +     +     +

| 1  1  1  1  1|80 80|79 79|78 78|

+              +--+--+--+--+--+--+

| 1  1  1  1  1|75 75|76 76|77 77|

+              +     +     +     +

| 1  1  1  1  1|75 75|76 76|77 77|

+              +--+--+--+--+--+--+

| 1  1  1  1  1|74 74|73 73|72 72|

+--+--+--+--+--+     +     +     +

|47 47|62 62 62|74 74|73 73|72 72|

+     +        +--+--+--+--+--+--+

|47 47|62 62 62|67 67 67|66 66 66|

+--+--+        +        +        +

|48 48|62 62 62|67 67 67|66 66 66|

+     +--+--+--+        +        +

|48 48|61 61 61|67 67 67|66 66 66|

+--+--+        +--+--+--+--+--+--+

|49 49|61 61 61|65 65|64 64|63 63|

+     +        +     +     +     +

|49 49|61 61 61|65 65|64 64|63 63|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

| 1  1  1  1  1|70 70 70|69 69 69|

+              +        +        +

| 1  1  1  1  1|70 70 70|69 69 69|

+              +        +        +

| 1  1  1  1  1|70 70 70|69 69 69|

+              +--+--+--+--+--+--+

| 1  1  1  1  1|71 71 71|68 68 68|

+              +        +        +

| 1  1  1  1  1|71 71 71|68 68 68|

+--+--+--+--+--+        +        +

|46 46|62 62 62|71 71 71|68 68 68|

+     +        +--+--+--+--+--+--+

|46 46|62 62 62|67 67 67|66 66 66|

+--+--+        +        +        +

|56 56|62 62 62|67 67 67|66 66 66|

+     +--+--+--+        +        +

|56 56|61 61 61|67 67 67|66 66 66|

+--+--+        +--+--+--+--+--+--+

|57 57|61 61 61|60 60|59 59|58 58|

+     +        +     +     +     +

|57 57|61 61 61|60 60|59 59|58 58|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

| 1  1  1  1  1|70 70 70|69 69 69|

+              +        +        +

| 1  1  1  1  1|70 70 70|69 69 69|

+              +        +        +

| 1  1  1  1  1|70 70 70|69 69 69|

+              +--+--+--+--+--+--+

| 1  1  1  1  1|71 71 71|68 68 68|

+              +        +        +

| 1  1  1  1  1|71 71 71|68 68 68|

+--+--+--+--+--+        +        +

|46 46|44 44 44|71 71 71|68 68 68|

+     +        +--+--+--+--+--+--+

|46 46|44 44 44|43 43 43|42 42 42|

+--+--+        +        +        +

|56 56|44 44 44|43 43 43|42 42 42|

+     +--+--+--+        +        +

|56 56|53 53 53|43 43 43|42 42 42|

+--+--+        +--+--+--+--+--+--+

|57 57|53 53 53|60 60|59 59|58 58|

+     +        +     +     +     +

|57 57|53 53 53|60 60|59 59|58 58|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

| 1  1  1  1  1|70 70 70|69 69 69|

+              +        +        +

| 1  1  1  1  1|70 70 70|69 69 69|

+              +        +        +

| 1  1  1  1  1|70 70 70|69 69 69|

+              +--+--+--+--+--+--+

| 1  1  1  1  1|71 71 71|68 68 68|

+              +        +        +

| 1  1  1  1  1|71 71 71|68 68 68|

+--+--+--+--+--+        +        +

|45 45|44 44 44|71 71 71|68 68 68|

+     +        +--+--+--+--+--+--+

|45 45|44 44 44|43 43 43|42 42 42|

+--+--+        +        +        +

|55 55|44 44 44|43 43 43|42 42 42|

+     +--+--+--+        +        +

|55 55|53 53 53|43 43 43|42 42 42|

+--+--+        +--+--+--+--+--+--+

|54 54|53 53 53|52 52|51 51|50 50|

+     +        +     +     +     +

|54 54|53 53 53|52 52|51 51|50 50|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

|39 39|40 40|41 41|25 25 25|28 28|

+     +     +     +        +     +

|39 39|40 40|41 41|25 25 25|28 28|

+--+--+--+--+--+--+        +--+--+

|38 38 38|37 37 37|25 25 25|27 27|

+        +        +--+--+--+     +

|38 38 38|37 37 37|24 24 24|27 27|

+        +        +        +--+--+

|38 38 38|37 37 37|24 24 24|26 26|

+--+--+--+--+--+--+        +     +

|45 45|44 44 44| 0|24 24 24|26 26|

+     +        +--+--+--+--+--+--+

|45 45|44 44 44|43 43 43|42 42 42|

+--+--+        +        +        +

|55 55|44 44 44|43 43 43|42 42 42|

+     +--+--+--+        +        +

|55 55|53 53 53|43 43 43|42 42 42|

+--+--+        +--+--+--+--+--+--+

|54 54|53 53 53|52 52|51 51|50 50|

+     +        +     +     +     +

|54 54|53 53 53|52 52|51 51|50 50|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

|39 39|40 40|41 41|25 25 25|28 28|

+     +     +     +        +     +

|39 39|40 40|41 41|25 25 25|28 28|

+--+--+--+--+--+--+        +--+--+

|38 38 38|37 37 37|25 25 25|27 27|

+        +        +--+--+--+     +

|38 38 38|37 37 37|24 24 24|27 27|

+        +        +        +--+--+

|38 38 38|37 37 37|24 24 24|26 26|

+--+--+--+--+--+--+        +     +

|15 15|14 14|13 13|24 24 24|26 26|

+     +     +     +--+--+--+--+--+

|15 15|14 14|13 13| 2  2  2  2  2|

+--+--+--+--+--+--+              +

|10 10|11 11|12 12| 2  2  2  2  2|

+     +     +     +              +

|10 10|11 11|12 12| 2  2  2  2  2|

+--+--+--+--+--+--+              +

| 9  9| 8  8| 7  7| 2  2  2  2  2|

+     +     +     +              +

| 9  9| 8  8| 7  7| 2  2  2  2  2|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

|36 36|35 35|34 34|25 25 25|23 23|

+     +     +     +        +     +

|36 36|35 35|34 34|25 25 25|23 23|

+--+--+--+--+--+--+        +--+--+

|38 38 38|37 37 37|25 25 25|22 22|

+        +        +--+--+--+     +

|38 38 38|37 37 37|24 24 24|22 22|

+        +        +        +--+--+

|38 38 38|37 37 37|24 24 24|21 21|

+--+--+--+--+--+--+        +     +

|15 15|14 14|13 13|24 24 24|21 21|

+     +     +     +--+--+--+--+--+

|15 15|14 14|13 13| 2  2  2  2  2|

+--+--+--+--+--+--+              +

|10 10|11 11|12 12| 2  2  2  2  2|

+     +     +     +              +

|10 10|11 11|12 12| 2  2  2  2  2|

+--+--+--+--+--+--+              +

| 9  9| 8  8| 7  7| 2  2  2  2  2|

+     +     +     +              +

| 9  9| 8  8| 7  7| 2  2  2  2  2|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

|36 36|35 35|34 34|20 20 20|23 23|

+     +     +     +        +     +

|36 36|35 35|34 34|20 20 20|23 23|

+--+--+--+--+--+--+        +--+--+

|30 30 30|29 29 29|20 20 20|22 22|

+        +        +--+--+--+     +

|30 30 30|29 29 29|17 17 17|22 22|

+        +        +        +--+--+

|30 30 30|29 29 29|17 17 17|21 21|

+--+--+--+--+--+--+        +     +

| 5  5  5| 6  6  6|17 17 17|21 21|

+        +        +--+--+--+--+--+

| 5  5  5| 6  6  6| 2  2  2  2  2|

+        +        +              +

| 5  5  5| 6  6  6| 2  2  2  2  2|

+--+--+--+--+--+--+              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+        +        +              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+        +        +              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

|31 31|32 32|33 33|20 20 20|19 19|

+     +     +     +        +     +

|31 31|32 32|33 33|20 20 20|19 19|

+--+--+--+--+--+--+        +--+--+

|30 30 30|29 29 29|20 20 20|18 18|

+        +        +--+--+--+     +

|30 30 30|29 29 29|17 17 17|18 18|

+        +        +        +--+--+

|30 30 30|29 29 29|17 17 17|16 16|

+--+--+--+--+--+--+        +     +

| 5  5  5| 6  6  6|17 17 17|16 16|

+        +        +--+--+--+--+--+

| 5  5  5| 6  6  6| 2  2  2  2  2|

+        +        +              +

| 5  5  5| 6  6  6| 2  2  2  2  2|

+--+--+--+--+--+--+              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+        +        +              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+        +        +              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+--+--+--+--+--+--+--+--+--+--+--+

+--+--+--+--+--+--+--+--+--+--+--+

|31 31|32 32|33 33|20 20 20|19 19|

+     +     +     +        +     +

|31 31|32 32|33 33|20 20 20|19 19|

+--+--+--+--+--+--+        +--+--+

|30 30 30|29 29 29|20 20 20|18 18|

+        +        +--+--+--+     +

|30 30 30|29 29 29|17 17 17|18 18|

+        +        +        +--+--+

|30 30 30|29 29 29|17 17 17|16 16|

+--+--+--+--+--+--+        +     +

| 5  5  5| 6  6  6|17 17 17|16 16|

+        +        +--+--+--+--+--+

| 5  5  5| 6  6  6| 2  2  2  2  2|

+        +        +              +

| 5  5  5| 6  6  6| 2  2  2  2  2|

+--+--+--+--+--+--+              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+        +        +              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+        +        +              +

| 4  4  4| 3  3  3| 2  2  2  2  2|

+--+--+--+--+--+--+--+--+--+--+--+

```

-------
So, for example, on the base, 5-cm cube 2 lies at the lower right. A 6x6 area lies to the left of that and at this level is a set of 4 3-cm cubes (cubes 3 thru 6), and if you follow that area up, you'll see it's overlain by a layer of nine 2-cm cubes (cubes 7 - 15). At the top, cube 1 is the 5-cm cube and 72-80 take up the 6x6 area (arbitrarily the 2-cm cubes are on top and the 3-cm cubes 68-71 lie below them, rather than above).

The empty unit cube is labeled 0.
 Posted by Charlie on 2004-01-04 20:17:38
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