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Rameses' Pyramid (Posted on 2004-11-05) Difficulty: 3 of 5
A classic:

Rameses wishes to build a great pyramid for his interment.

The structure will have a square base and be solidly composed of cubical stone blocks. Each level of the pyramid contains one block fewer per side as the pyramid rises.

Rameses has available an initial work force of 35,000 slaves. Each morning the available labor pool is divided into work crews of 17 slaves each. Any remainder that cannot form a full crew gets the day off but are available the following day. Each crew can lay one block of the pyramid each day.

Unfortunately, the heat of the desert sun causes the death of one member of each crew each day. Work ceases on the project when it can be determined that there will be insufficient slaves available to raise the pyramid one more level. Each stone block measures 3 meters per side.

How many days will it take to construct Rameses' pyramid? How tall will it be? How many of the original slaves survive the construction?

See The Solution Submitted by SilverKnight    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 3 of 15 |

If you continue until you have as many blocks as possible, you would stop when there are fewer than 17 slaves left, so there would be 16 left at the end.  One block is laid for every slave killed, so 35,000 - 16 = 34,984 blocks could be laid.  The number of days is harder. At the end, we have to see how high the pyramid will be with 34,984 blocks, see how many are really needed, and adjust the slave count to that.

An approximation to the number of days could be made by stating that n = 35000 e^(-t/17), so that dn/dt = -n/17, and solving for n = 1, which would come out to 129.7... days, rounded to 130 days.  But the number of crews is not a continuous function, and the approximation is 6 days off, based on the following computer-generated result:

On the first day there are 2058 crews, laying 2058 blocks, leaving 32,942 workers for the next day.  The second day there are 1937 crews, bringing the total number of blocks laid thus far to 3995 and leaving 31,005 workers for the next day.  This continues as in the following table:

  1  2058  2058 32942
  2  1937  3995 31005
  3  1823  5818 29182
  4  1716  7534 27466
  5  1615  9149 25851
  6  1520 10669 24331
  7  1431 12100 22900
  8  1347 13447 21553
  9  1267 14714 20286
 10  1193 15907 19093
 11  1123 17030 17970
 12  1057 18087 16913
 13   994 19081 15919
 14   936 20017 14983
 15   881 20898 14102
 16   829 21727 13273
 17   780 22507 12493
 18   734 23241 11759
 19   691 23932 11068
 20   651 24583 10417
 21   612 25195  9805
 22   576 25771  9229
 23   542 26313  8687
 24   511 26824  8176
 25   480 27304  7696
 26   452 27756  7244
 27   426 28182  6818
 28   401 28583  6417
 29   377 28960  6040
 30   355 29315  5685
 31   334 29649  5351
 32   314 29963  5037
 33   296 30259  4741
 34   278 30537  4463
 35   262 30799  4201
 36   247 31046  3954
 37   232 31278  3722
 38   218 31496  3504
 39   206 31702  3298
 40   194 31896  3104
 41   182 32078  2922
 42   171 32249  2751
 43   161 32410  2590
 44   152 32562  2438
 45   143 32705  2295
 46   135 32840  2160
 47   127 32967  2033
 48   119 33086  1914
 49   112 33198  1802
 50   106 33304  1696
 51    99 33403  1597
 52    93 33496  1504
 53    88 33584  1416
 54    83 33667  1333
 55    78 33745  1255
 56    73 33818  1182
 57    69 33887  1113
 58    65 33952  1048
 59    61 34013   987
 60    58 34071   929
 61    54 34125   875
 62    51 34176   824
 63    48 34224   776
 64    45 34269   731
 65    43 34312   688
 66    40 34352   648
 67    38 34390   610
 68    35 34425   575
 69    33 34458   542
 70    31 34489   511
 71    30 34519   481
 72    28 34547   453
 73    26 34573   427
 74    25 34598   402
 75    23 34621   379
 76    22 34643   357
 77    21 34664   336
 78    19 34683   317
 79    18 34701   299
 80    17 34718   282
 81    16 34734   266
 82    15 34749   251
 83    14 34763   237
 84    13 34776   224
 85    13 34789   211
 86    12 34801   199
 87    11 34812   188
 88    11 34823   177
 89    10 34833   167
 90     9 34842   158
 91     9 34851   149
 92     8 34859   141
 93     8 34867   133
 94     7 34874   126
 95     7 34881   119
 96     7 34888   112
 97     6 34894   106
 98     6 34900   100
 99     5 34905    95
100     5 34910    90
101     5 34915    85
102     5 34920    80
103     4 34924    76
104     4 34928    72
105     4 34932    68
106     4 34936    64
107     3 34939    61
108     3 34942    58
109     3 34945    55
110     3 34948    52
111     3 34951    49
112     2 34953    47
113     2 34955    45
114     2 34957    43
115     2 34959    41
116     2 34961    39
117     2 34963    37
118     2 34965    35
119     2 34967    33
120     1 34968    32
121     1 34969    31
122     1 34970    30
123     1 34971    29
124     1 34972    28
125     1 34973    27
126     1 34974    26
127     1 34975    25
128     1 34976    24
129     1 34977    23
130     1 34978    22
131     1 34979    21
132     1 34980    20
133     1 34981    19
134     1 34982    18
135     1 34983    17
136     1 34984    16

So at the end of the 136th day, 34,984 blocks will have been laid, with 16 slaves remaining. This assumes we lay all 34,984 possible.

This sequence was produced by

n = 35000
OPEN "ramsespr.txt" FOR OUTPUT AS #2
DO
  day = day + 1
  crews = INT(n / 17)
  blocks = blocks + crews
  n = n - crews
  PRINT #2, USING "### ##### ##### #####"; day; crews; blocks; n
LOOP UNTIL n < 17

For the height of the pyramid (and a shortening of the time and number of slaves used):

The top level has 1 block; the second level has 4; the third, 9; etc. -- the square numbers.

A table, counting rows from the top of blocks used in each row, and total blocks at that level and higher  shows:

1             1             1
2             4             5
3             9             14
4             16            30
5             25            55
6             36            91
7             49            140
8             64            204
9             81            285
10            100           385
11            121           506
12            144           650
13            169           819
14            196           1015
15            225           1240
16            256           1496
17            289           1785
18            324           2109
19            361           2470
20            400           2870
21            441           3311
22            484           3795
23            529           4324
24            576           4900
25            625           5525
26            676           6201
27            729           6930
28            784           7714
29            841           8555
30            900           9455
31            961           10416
32            1024          11440
33            1089          12529
34            1156          13685
35            1225          14910
36            1296          16206
37            1369          17575
38            1444          19019
39            1521          20540
40            1600          22140
41            1681          23821
42            1764          25585
43            1849          27434
44            1936          29370
45            2025          31395
46            2116          33511
47            2209          35720

so only 33,511 blocks will actually be needed, so actually, 35,000 - 33,511 = 1,489 slaves can be spared death.  From the above table, we can see that this can be reached on the 53rd day by sending out only 15 crews that day, instead of the 88 available.

The pyramid will be 46 courses high and so is 46*3 = 138 meters high.


  Posted by Charlie on 2004-11-05 09:32:20
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