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 Never 0 mod 3 (Posted on 2016-03-17)
Let k be a positive integer. Suppose that the integers 1, 2, 3, ...3k, 3k + 1 are written down in random order.

What is the probability that at no time during this process, the sum of the integers that have been written up to that time is divisible by 3?

Source: Putnam competition

 No Solution Yet Submitted by Ady TZIDON No Rating

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 re(3): Solution | Comment 5 of 7 |
(In reply to re(2): Solution by Charlie)

I've recreated Brian's logic and it seems impeccable.  Yet there are 360 valid cases out of 5040 permutations of 1234567:

1342675
1342756
1342765
1345672
1345726
1345762
1346275
1346572
1364275
1364572
1367245
1367542
1372456
1372465
1372645
1375426
1375462
1375642
1376245
1376542
1423675
1423756
1423765
1426375
1426735
1426753
1427356
1427365
1427536
1427563
1427635
1427653
1432675
1432756
1432765
1435672
1435726
1435762
1436275
1436572
1453672
1453726
1453762
1456372
1456723
1456732
1457236
1457263
1457326
1457362
1457623
1457632
1462375
1462735
1462753
1463275
1463572
1465372
1465723
1465732
1634275
1634572
1637245
1637542
1642375
1642735
1642753
1643275
1643572
1645372
1645723
1645732
1672345
1672435
1672453
1673245
1673542
1675342
1675423
1675432
1723456
1723465
1723645
1724356
1724365
1724536
1724563
1724635
1724653
1726345
1726435
1726453
1732456
1732465
1732645
1735426
1735462
1735642
1736245
1736542
1753426
1753462
1753642
1754236
1754263
1754326
1754362
1754623
1754632
1756342
1756423
1756432
1762345
1762435
1762453
1763245
1763542
1765342
1765423
1765432
4123675
4123756
4123765
4126375
4126735
4126753
4127356
4127365
4127536
4127563
4127635
4127653
4132675
4132756
4132765
4135672
4135726
4135762
4136275
4136572
4153672
4153726
4153762
4156372
4156723
4156732
4157236
4157263
4157326
4157362
4157623
4157632
4162375
4162735
4162753
4163275
4163572
4165372
4165723
4165732
4312675
4312756
4312765
4315672
4315726
4315762
4316275
4316572
4361275
4361572
4367215
4367512
4372156
4372165
4372615
4375126
4375162
4375612
4376215
4376512
4612375
4612735
4612753
4613275
4613572
4615372
4615723
4615732
4631275
4631572
4637215
4637512
4672135
4672153
4672315
4673215
4673512
4675123
4675132
4675312
4721356
4721365
4721536
4721563
4721635
4721653
4723156
4723165
4723615
4726135
4726153
4726315
4732156
4732165
4732615
4735126
4735162
4735612
4736215
4736512
4751236
4751263
4751326
4751362
4751623
4751632
4753126
4753162
4753612
4756123
4756132
4756312
4762135
4762153
4762315
4763215
4763512
4765123
4765132
4765312
7123456
7123465
7123645
7124356
7124365
7124536
7124563
7124635
7124653
7126345
7126435
7126453
7132456
7132465
7132645
7135426
7135462
7135642
7136245
7136542
7153426
7153462
7153642
7154236
7154263
7154326
7154362
7154623
7154632
7156342
7156423
7156432
7162345
7162435
7162453
7163245
7163542
7165342
7165423
7165432
7312456
7312465
7312645
7315426
7315462
7315642
7316245
7316542
7342156
7342165
7342615
7345126
7345162
7345612
7346215
7346512
7361245
7361542
7364215
7364512
7421356
7421365
7421536
7421563
7421635
7421653
7423156
7423165
7423615
7426135
7426153
7426315
7432156
7432165
7432615
7435126
7435162
7435612
7436215
7436512
7451236
7451263
7451326
7451362
7451623
7451632
7453126
7453162
7453612
7456123
7456132
7456312
7462135
7462153
7462315
7463215
7463512
7465123
7465132
7465312
7612345
7612435
7612453
7613245
7613542
7615342
7615423
7615432
7631245
7631542
7634215
7634512
7642135
7642153
7642315
7643215
7643512
7645123
7645132
7645312

 Posted by Charlie on 2016-03-17 22:33:33

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