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#1 2006-03-05 03:40:44

Ansette
Member
Registered: 2006-02-19
Posts: 21

Pure problem

We've been givin a problem involving boxes (parcels basically)..

The dimensions of each parcel are integers abc with 1≤a≤b≤c≤n

Basically such that for each n, there are many combinations as you can see, for instance with n=2, the options are:

1,1,1
1,1,2
1,2,2
2,2,2

to fit within those bounds.

The first thing we've noticed is that the number of parcels per each n is a tetrahedral number.. e.g. general formula with our n = n(n+1)(n+2)/6

e.g. 1,4,10,20,35.. etc

The next thing of note is that the total volume for each set of parcels fits the pattern of is 1,15,90,350,1050.. Stirling numbers of second order so i'm informed by the internet..

What I'm trying to ask is for an explanation as to why this is the case?
(And also a general formula for stirling numbers of second order if possible!)

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#2 2006-03-05 04:19:12

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

First look at 2d.
It will help you with the 3d.


IPBLE:  Increasing Performance By Lowering Expectations.

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#3 2006-03-05 04:27:10

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

Let


is the number of all sets of k elements

for which

Last edited by krassi_holmz (2006-03-05 04:28:43)


IPBLE:  Increasing Performance By Lowering Expectations.

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#4 2006-03-05 04:31:05

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

How can we compute

?
I noticed this:


IPBLE:  Increasing Performance By Lowering Expectations.

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#5 2006-03-05 04:39:06

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

Cool!!!
Here's a proof:
for some x the number of all sets

--->(1)
for which

is exactly

If we summarize (1) for all x≤n then we'll get the number or all sets for which

which is exacly


IPBLE:  Increasing Performance By Lowering Expectations.

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#6 2006-03-05 04:46:11

Ansette
Member
Registered: 2006-02-19
Posts: 21

Re: Pure problem

i have to admit i'm slightly confused..

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#7 2006-03-05 04:59:09

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

It's not hard to consider that

.
So


so here's your explanation why the number is tetrahedral number.


IPBLE:  Increasing Performance By Lowering Expectations.

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#8 2006-03-05 05:07:49

Ansette
Member
Registered: 2006-02-19
Posts: 21

Re: Pure problem

ah of course.. working like the binomial theorem..!!

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#9 2006-03-05 05:13:43

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

It can be shown with simple induction that:


IPBLE:  Increasing Performance By Lowering Expectations.

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#10 2006-03-05 05:16:23

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

This was the complete examination of your first problem.
The second seems much interesting for me.
Now I'll search the web for these stirling numbers.


IPBLE:  Increasing Performance By Lowering Expectations.

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#11 2006-03-05 05:32:30

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

After some web search I can tell you that if we name your sequence with a:
a1=1
a2=15
a3=90
a4=350
a5=1050

It seems that:


IPBLE:  Increasing Performance By Lowering Expectations.

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#12 2006-03-05 05:39:26

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

With some simplification it seems that the formula is:

, where
is the binomial coeficient.

Last edited by krassi_holmz (2006-03-05 05:42:09)


IPBLE:  Increasing Performance By Lowering Expectations.

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#13 2006-03-05 05:46:31

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

I tried and it seems that the above formula is correct.
I don't know why but it's true.
It will be hard task to simplify it more.
That was very good question!!!


IPBLE:  Increasing Performance By Lowering Expectations.

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#14 2006-03-05 06:04:47

Ansette
Member
Registered: 2006-02-19
Posts: 21

Re: Pure problem

ok im very impressed lol.. i had some trouble making the formula work at first.. misread the last j as a 3 so was writing iC3 each time.. really math it up lol..

there is a third question really..

we have to try and place these individual boxes into a large parcel that must be rectangular with no free space, though the dimensions are irrelevant.. kind of like a three-dimensional tetris..

any thoughts?

(alternatively if one parcel is not possible, as few as possible..)

obviously for n=1, 1 package
for n=2 it requires 2, but then that's understandable with a volume of 15.
for n=3 i believe we've managed 1, but after that we're unsure

Last edited by Ansette (2006-03-05 06:07:16)

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#15 2006-03-05 06:17:16

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

Good.
Can I ask where are this questions from?
There are very interesting but...
very hard too.
The third is the hardest.
My suggestion: try solving it in 2d first. I'll try too.
I'll have a closer look at a_i.


IPBLE:  Increasing Performance By Lowering Expectations.

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#16 2006-03-05 06:18:55

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

Here are the first 50 terms of a_i:
{1, 15, 90, 350, 1050, 2646, 5880, 11880, 22275, 39325, 66066, 106470,
165620, 249900, 367200, 527136, 741285, 1023435, 1389850, 1859550, 2454606,
3200450, 4126200, 5265000, 6654375, 8336601, 10359090, 12774790, 15642600,
19027800, 23002496, 27646080, 33045705, 39296775, 46503450, 54779166,
64247170, 75041070, 87305400, 101196200, 116881611, 134542485, 154373010,
176581350, 201390300, 229037956, 259778400, 293882400, 331638125, 373351875}


IPBLE:  Increasing Performance By Lowering Expectations.

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#17 2006-03-05 06:24:36

Ansette
Member
Registered: 2006-02-19
Posts: 21

Re: Pure problem

we have a presentation on this, first year group work basically (designed no doubt in a way to build team skills..) we'd noticed a lot of the patterns between us however were unsure what to make of them or how to prove them :s

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#18 2006-03-05 06:26:01

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

And here's a set of the divisors of each (you may need it smile ):
{{1}, {1, 3,
   5, 15}, {1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90}, {1, 2, 5, 7, 10, 14,
    25, 35, 50, 70, 175, 350}, {1, 2, 3, 5, 6,
   7, 10, 14, 15, 21, 25, 30, 35, 42, 50, 70, 75, 105, 150, 175, 210, 350,
    525, 1050}, {1, 2, 3, 6,
  7, 9, 14, 18, 21, 27, 42, 49, 54, 63, 98, 126, 147, 189, 294, 378, 441, 882,
     1323, 2646}, {1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28,
     30, 35, 40, 42, 49, 56, 60, 70, 84, 98, 105, 120, 140, 147, 168, 196,
    210, 245, 280, 294, 392, 420, 490, 588, 735, 840, 980, 1176, 1470,
    1960, 2940, 5880}, {1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 18,
    20, 22, 24, 27, 30, 33, 36, 40, 44, 45, 54, 55, 60, 66, 72, 88, 90, 99,
    108, 110, 120, 132, 135, 165, 180, 198, 216, 220, 264, 270, 297, 330, 360,
     396, 440, 495, 540, 594, 660, 792, 990, 1080, 1188, 1320, 1485, 1980,
    2376, 2970, 3960, 5940, 11880}, {1, 3, 5, 9, 11, 15, 25, 27, 33, 45, 55,
    75, 81, 99, 135, 165, 225, 275, 297, 405, 495, 675, 825, 891,
    1485, 2025, 2475, 4455, 7425, 22275}, {1, 5, 11, 13, 25, 55, 65,
    121, 143, 275, 325, 605, 715, 1573, 3025, 3575, 7865, 39325}, {1, 2, 3, 6,
     7, 11, 13, 14,
   21, 22, 26, 33, 39, 42, 66, 77, 78, 91, 121, 143, 154, 182, 231, 242,
    273, 286, 363, 429, 462, 546, 726, 847, 858, 1001, 1573, 1694, 2002, 2541,
     3003, 3146, 4719, 5082, 6006, 9438, 11011, 22022, 33033, 66066}, {1, 2,
    3, 5, 6, 7, 9, 10, 13, 14, 15, 18, 21, 26, 30, 35, 39, 42, 45, 63, 65,
     70, 78, 90, 91, 105, 117, 126, 130, 169, 182, 195, 210, 234, 273, 315,
    338, 390, 455, 507, 546, 585, 630, 819, 845, 910, 1014, 1170, 1183, 1365,
    1521, 1638,
    1690, 2366, 2535, 2730, 3042, 3549, 4095, 5070, 5915, 7098, 7605, 8190,
10647, 11830, 15210, 17745, 21294, 35490, 53235, 106470}, {1, 2, 4, 5, 7,
    10, 13, 14, 20, 26, 28, 35, 49, 52, 65, 70, 91, 98, 130, 140, 169, 182,
    196, 245, 260, 338, 364, 455, 490, 637, 676, 845, 910, 980, 1183, 1274,
    1690, 1820, 2366, 2548, 3185, 3380, 4732, 5915, 6370, 8281, 11830, 12740,
    16562, 23660, 33124, 41405, 82810,
   165620}, {1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 17, 20, 21, 25, 28, 30, 34,
    35, 42, 49, 50, 51, 60, 68, 70, 75, 84, 85, 98, 100, 102, 105, 119,
    140, 147, 150, 170, 175, 196, 204, 210, 238, 245, 255, 294, 300, 340,
    350, 357, 420, 425, 476, 490, 510, 525, 588, 595, 700, 714, 735, 833,
    850, 980, 1020, 1050, 1190, 1225, 1275, 1428, 1470, 1666, 1700, 1785,
    2100, 2380, 2450, 2499, 2550, 2940, 2975, 3332, 3570, 3675,
    4165, 4900, 4998, 5100, 5950, 7140, 7350, 8330, 8925, 9996, 11900, 12495,
    14700, 16660, 17850, 20825, 24990, 35700, 41650, 49980, 62475, 83300,
    124950, 249900}, {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17, 18, 20, 24,
    25, 27, 30, 32, 34, 36, 40, 45, 48, 50, 51, 54, 60, 68, 72, 75, 80, 85,
    90, 96, 100, 102, 108, 120, 135, 136, 144, 150, 153, 160, 170, 180, 200,
     204, 216, 225, 240, 255, 270, 272, 288, 300, 306, 340, 360, 400, 408, 425
    , 432, 450, 459, 480, 510, 540, 544, 600, 612, 675, 680, 720, 765,
     800, 816, 850, 864, 900, 918, 1020, 1080, 1200, 1224, 1275, 1350, 1360,
     1440, 1530, 1632, 1700, 1800, 1836, 2040, 2160, 2295, 2400, 2448, 2550,
    2700, 2720, 3060, 3400, 3600, 3672, 3825, 4080, 4320, 4590, 4896, 5100,
     5400, 6120, 6800, 7200, 7344, 7650, 8160, 9180, 10200, 10800, 11475,
    12240, 13600, 14688, 15300, 18360, 20400, 21600, 22950, 24480, 30600,
36720, 40800, 45900, 61200, 73440, 91800, 122400, 183600, 367200}, {
    1, 2, 3, 4, 6, 8, 12, 16, 17, 19, 24, 32, 34, 38, 48, 51, 57, 68, 76,
    96, 102, 114, 136, 152, 204, 228, 272, 289, 304, 323, 408, 456, 544,
    578, 608, 646, 816, 867, 912, 969, 1156, 1292, 1632, 1734, 1824, 1938,
    2312, 2584, 3468, 3876, 4624, 5168, 5491, 6936, 7752, 9248, 10336, 10982,
    13872, 15504, 16473, 21964, 27744, 31008, 32946, 43928, 65892,
    87856, 131784, 175712, 263568, 527136}, {1, 3, 5, 9, 15, 17, 19, 27, 45,
    51, 57, 85, 95, 135, 153, 171, 255, 285, 289, 323, 459, 513, 765,
    855, 867, 969, 1445, 1615, 2295, 2565, 2601, 2907, 4335, 4845, 5491, 7803,
     8721, 13005, 14535, 16473, 27455, 39015, 43605, 49419, 82365, 148257,
    247095, 741285}, {1, 3, 5, 7, 9, 15, 19, 21, 27, 35, 45, 57, 63, 81,
    95, 105, 133, 135, 171, 189, 285, 315, 361, 399, 405, 513, 567, 665,
    855, 945, 1083, 1197, 1539, 1805, 1995, 2527, 2565, 2835, 3249,
    3591, 5415, 5985, 7581, 7695, 9747, 10773, 12635, 16245, 17955, 22743,
    29241, 37905, 48735, 53865, 68229, 113715, 146205, 204687,
     341145, 1023435}, {1, 2, 5, 7, 10, 11, 14, 19, 22, 25, 35, 38, 50,
    55, 70, 77, 95, 110, 133, 154, 175, 190, 209, 266, 275, 350,
    361, 385, 418, 475, 550, 665, 722, 770,
   950, 1045, 1330, 1463, 1805, 1925, 2090, 2527, 2926, 3325, 3610, 3850,
    3971, 5054, 5225, 6650, 7315, 7942, 9025, 10450, 12635,
    14630, 18050, 19855, 25270, 27797, 36575, 39710, 55594, 63175, 73150,
    99275, 126350, 138985, 198550, 277970, 694925, 1389850}, {1,
    2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 23, 25, 30, 33, 35, 42,
    46, 49, 50, 55, 66, 69, 70, 75, 77, 98, 105, 110, 115, 138, 147, 150,
  154, 161, 165, 175, 210, 230, 231, 245, 253, 275, 294, 322, 330, 345, 350,
    385, 462, 483, 490, 506, 525, 539, 550, 575, 690, 735, 759, 770, 805, 825,
     966, 1050, 1078, 1127, 1150, 1155, 1225, 1265, 1470, 1518, 1610,
    1617, 1650, 1725, 1771, 1925, 2254, 2310, 2415, 2450, 2530, 2695, 3234,
    3381, 3450, 3542, 3675, 3795, 3850, 4025, 4830, 5313, 5390, 5635, 5775,
    6325, 6762, 7350, 7590, 8050, 8085, 8855, 10626, 11270, 11550, 12075,
    12397, 12650, 13475, 16170,
     16905, 17710, 18975, 24150, 24794, 26565, 26950, 28175, 33810, 37191,
37950, 40425, 44275, 53130, 56350, 61985, 74382, 80850, 84525, 88550, 123970,
    132825, 169050, 185955, 265650, 309925, 371910, 619850,
    929775, 1859550}, {1, 2, 3, 6, 7, 9, 11, 14, 18, 21, 22, 23, 33, 42, 46,
    49, 63, 66, 69, 77, 98, 99, 121, 126, 138, 147, 154, 161, 198,
     207, 231, 242, 253, 294, 322, 363, 414, 441, 462, 483, 506, 539, 693,
    726, 759, 847, 882, 966, 1078, 1089, 1127, 1386, 1449, 1518, 1617, 1694,
    1771, 2178, 2254, 2277, 2541, 2783, 2898, 3234,
    3381, 3542, 4554, 4851, 5082, 5313,
  5566, 5929, 6762, 7623, 8349, 9702, 10143, 10626, 11858, 12397, 15246,
     15939, 16698, 17787, 19481, 20286, 24794, 25047, 31878, 35574, 37191,
    38962, 50094, 53361, 58443, 74382, 106722, 111573, 116886, 136367, 175329,
     223146, 272734, 350658, 409101, 818202, 1227303, 2454606}, {1, 2, 5,
    10, 11, 22, 23, 25, 46, 50, 55, 110, 115, 121, 230, 242, 253, 275, 506,
    529, 550, 575, 605, 1058, 1150, 1210, 1265, 2530, 2645, 2783, 3025, 5290,
    5566, 5819, 6050, 6325, 11638, 12650, 13225, 13915, 26450, 27830, 29095,
    58190, 64009, 69575, 128018, 139150, 145475, 290950, 320045, 640090,
    1600225, 3200450}, {1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 20, 23, 24, 25,
    26, 30, 39, 40, 46, 50, 52, 60, 65, 69, 75, 78, 92, 100, 104, 115, 120,
    130, 138, 150, 156, 184, 195, 200, 230, 260, 276, 299, 300, 312, 325, 345,
     390, 460, 520, 529, 552, 575, 598, 600, 650, 690, 780, 897, 920, 975,
    1058, 1150, 1196, 1300, 1380, 1495, 1560, 1587, 1725, 1794, 1950, 2116,
     2300, 2392, 2600, 2645, 2760, 2990, 3174, 3450, 3588, 3900, 4232, 4485,
     4600, 5290, 5980, 6348, 6877, 6900, 7176, 7475, 7800, 7935, 8970, 10580,
    11960, 12696, 13225, 13754, 13800, 14950, 15870, 17940, 20631, 21160,
    22425, 26450, 27508, 29900, 31740, 34385, 35880, 39675, 41262, 44850,
    52900, 55016, 59800, 63480, 68770, 79350, 82524, 89700, 103155, 105800,
    137540, 158700, 165048, 171925, 179400, 206310,
    275080, 317400, 343850, 412620, 515775, 687700, 825240, 1031550, 1375400,
2063100, 4126200}, {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 18, 20, 24,
    25, 26, 27, 30, 36, 39, 40, 45, 50, 52, 54, 60, 65, 72, 75, 78, 81, 90,
    100, 104, 108, 117, 120, 125, 130, 135, 150, 156, 162, 180, 195,
    200, 216, 225, 234, 250, 260, 270, 300, 312, 324, 325, 351, 360, 375,
    390, 405, 450, 468, 500, 520, 540, 585, 600, 625, 648, 650, 675, 702, 750,
     780, 810, 900, 936, 975, 1000, 1053, 1080, 1125, 1170, 1250, 1300,
    1350, 1404, 1500, 1560, 1620, 1625, 1755, 1800, 1875, 1950, 2025, 2106,
    2250, 2340, 2500, 2600, 2700, 2808, 2925, 3000, 3240, 3250, 3375, 3510,
     3750, 3900, 4050, 4212, 4500, 4680, 4875, 5000, 5265, 5400, 5625,
    5850, 6500, 6750, 7020, 7500, 7800, 8100, 8125, 8424, 8775, 9000, 9750,
    10125, 10530, 11250, 11700, 13000, 13500, 14040, 14625, 15000, 16200,
    16250, 16875, 17550, 19500, 20250, 21060, 22500, 23400, 24375,
    26325, 27000, 29250, 32500, 33750, 35100, 39000, 40500, 42120,
     43875, 45000, 48750, 50625, 52650, 58500, 65000, 67500, 70200, 73125,
    81000, 87750, 97500, 101250,
    105300, 117000, 131625, 135000, 146250, 175500, 195000, 202500, 210600,
219375, 263250, 292500, 351000, 405000, 438750, 526500, 585000, 658125,
877500, 1053000, 1316250, 1755000, 2632500, 5265000}, {1, 3, 5, 7, 9, 13, 15,
    21, 25, 35, 39, 45, 63, 65, 75, 91, 105, 117, 125, 169, 175,
    195, 225, 273, 315, 325, 375, 455, 507, 525, 585, 625, 819, 845, 875,
     975, 1125, 1183, 1365, 1521, 1575, 1625, 1875, 2275,
    2535, 2625, 2925, 3549, 4095, 4225, 4375, 4875, 5625, 5915, 6825,
    7605, 7875, 8125, 10647, 11375, 12675, 13125, 14625, 17745, 20475, 21125,
    24375, 29575, 34125, 38025, 39375, 53235, 56875, 63375, 73125, 88725,
    102375, 105625, 147875, 170625, 190125, 266175,
    316875, 443625, 511875, 739375, 950625, 1330875, 2218125, 6654375}, {1,
    3, 7, 9, 13, 21, 27, 29, 39, 63, 81, 87, 91, 117, 169, 189, 203, 243,
    261, 273, 351, 377, 507, 567, 609, 783, 819, 1053, 1131, 1183, 1521,
    1701, 1827, 2349, 2457, 2639, 3159, 3393, 3549, 4563, 4901, 5481, 7047,
    7371, 7917, 10179, 10647, 13689, 14703, 16443, 22113, 23751, 30537, 31941,
     34307, 41067, 44109, 49329, 71253, 91611, 95823, 102921,
     132327, 213759, 287469, 308763, 396981, 641277, 926289,
     1190943, 2778867, 8336601}, {1, 2, 3, 5, 6, 7, 9, 10, 14, 15,
     18, 21, 27, 29, 30, 35, 42, 45, 49, 54, 58, 63, 70, 81, 87, 90,
     98, 105, 126, 135, 145, 147, 162, 174, 189, 203,
    210, 243, 245, 261, 270, 290, 294, 315, 378, 405, 406, 435, 441, 486, 490,
     522, 567, 609, 630, 729, 735, 783, 810, 870, 882, 945, 1015,
    1134, 1215, 1218, 1305, 1323, 1421, 1458, 1470, 1566, 1701, 1827, 1890,
    2030, 2205, 2349, 2430, 2610, 2646, 2835, 2842, 3045, 3402, 3645,
    3654, 3915, 3969, 4263, 4410, 4698, 5103, 5481, 5670, 6090, 6615, 7047,
    7105, 7290, 7830, 7938, 8505, 8526, 9135, 10206, 10962, 11745, 11907,
    12789, 13230, 14094, 14210, 16443, 17010, 18270, 19845, 21141, 21315,
    23490, 23814, 25515, 25578, 27405, 32886, 35235, 35721, 38367, 39690,
    42282, 42630, 49329, 51030, 54810, 59535, 63945, 70470,
    71442, 76734, 82215, 98658, 105705, 115101, 119070, 127890, 147987,
164430, 178605, 191835, 211410, 230202, 246645, 295974, 345303, 357210,
383670, 493290, 575505, 690606, 739935, 1035909, 1151010, 1479870, 1726515,
    2071818, 3453030, 5179545, 10359090}, {1, 2, 5, 7, 10, 14, 29, 31,
     35, 49, 58, 62, 70, 98, 145, 155, 203, 217, 245, 290, 310, 406, 434,
    490, 841, 899, 1015, 1085, 1421, 1519, 1682, 1798, 2030, 2170, 2842, 3038,
     4205, 4495, 5887, 6293, 7105, 7595, 8410, 8990,
    11774, 12586, 14210, 15190, 26071, 29435, 31465, 41209, 44051, 52142,
    58870, 62930, 82418, 88102, 130355, 182497, 206045, 220255,
    260710, 364994, 412090, 440510, 912485, 1277479, 1824970, 2554958,
    6387395, 12774790}, {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 29, 30,
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     150, 155, 165, 180, 186, 198, 200, 220, 225, 248, 264, 275,
    279, 300, 310, 330, 341, 360, 372, 396, 440, 450, 465, 495, 550, 558,
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    165, 168, 170, 176, 187, 192, 204, 210, 220, 224, 231, 238, 240, 242, 255,
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IPBLE:  Increasing Performance By Lowering Expectations.

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#19 2006-03-05 06:33:45

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

Ansette wrote:

for n=3 i believe we've managed 1, but after that we're unsure

Who "we"?


IPBLE:  Increasing Performance By Lowering Expectations.

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#20 2006-03-05 07:41:21

Ansette
Member
Registered: 2006-02-19
Posts: 21

Re: Pure problem

group of students for this work..

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#21 2006-03-05 07:56:14

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

OK. Just don't forget putting my name in your presentation smile
I don't actually think I can handle with the 3rd problem.
Can you tell me more of that patterns?


IPBLE:  Increasing Performance By Lowering Expectations.

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#22 2006-03-05 11:39:26

Ansette
Member
Registered: 2006-02-19
Posts: 21

Re: Pure problem

well so far we have (from n=1)

1,2,1,4,5..

but then we're not even certain these are right!

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#23 2006-03-05 19:39:38

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

I have an idea.
I think I can prove your question number2 using the same methods as prob1:
Let

detones the sum of all products of k numbers
for which:
.
We can write this function like this:

Understand?
Clearly

Last edited by krassi_holmz (2006-03-05 19:48:47)


IPBLE:  Increasing Performance By Lowering Expectations.

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#24 2006-03-05 19:41:14

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

So


IPBLE:  Increasing Performance By Lowering Expectations.

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#25 2006-03-05 20:08:46

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Pure problem

OK. I'm done.
Here's the functional equation(I leave the froof to you):


So:

and, the main:


IPBLE:  Increasing Performance By Lowering Expectations.

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