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.
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.
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Posted by Charlie
on 2004-01-04 20:17:38 |
solution
