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Consecutive numbers

Math brain teasers require computations to solve.

 

Puzzle ID:#14553
Fun:*** (2.72)
Difficulty:*** (2.73)
Category:Math
Submitted By:Gerd*de*****

 

 

 



Between 1000 and 2000 you can get each integer as the sum of nonnegative consecutive integers. For example,

147+148+149+150+151+152+153 = 1050

There is only one number that you cannot get.

What is this number?

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Comments

krishnanAin*
Sep 01, 2003

Great teaser. Any multiple of an odd number is reachable and that leaves only the powers of two.
pi202**
Sep 08, 2003

A really fun teaser, but like a lot of fun teasers, not too difficult if you think about it in the right way.
heyevpos
Jan 19, 2004

Thank you, very well done, a great teaching tool.
javaguru*us*
Mar 11, 2009

Yes, very cool!

I like how that one fell into place with a little thought. Any odd number can be formed by adding two consecutive integers, and any value x * y where x is odd can be formed by centering x consecutive integers on y.

That left only even numbers with no odd factors, which are of course the powers of two.

Jimmygu3
Feb 26, 2014

I believe this answer, which has stood unchallenged for over 10 years, is incorrect.

Because the consecutive numbers must be nonnegative, a double prime like 1006 won't work. As javaguru stated, we would need to center 503 consecutive digits on 2. That string would begin with -249, which is not allowed. Extending javaguru's analysis, we see that x-1≤y, where x is the lowest odd factor. Any power of 2 (y) multiplied by a prime (x) that is greater than 2y+1, is a solution.

If you think I'm wrong, let me know! I'm seeing 150 additional solutions.

2*prime solutions:
1006
1018
1042
1046
1082
1094
1114
1126
1138
1142
1154
1174
1186
1198
1202
1214
1226
1234
1238
1262
1282
1286
1294
1306
1318
1322
1346
1354
1366
1382
1402
1418
1438
1454
1466
1478
1486
1502
1514
1522
1538
1546
1574
1594
1618
1622
1642
1646
1654
1658
1678
1706
1714
1718
1726
1754
1762
1766
1774
1814
1822
1838
1858
1874
1882
1894
1906
1934
1942
1954
1966
1982
1994
4*prime solutions:
1004
1028
1052
1076
1084
1108
1124
1132
1172
1228
1244
1252
1268
1324
1348
1388
1396
1412
1436
1468
1492
1516
1532
1556
1588
1604
1636
1676
1684
1724
1732
1756
1772
1796
1828
1844
1852
1868
1916
1948
1964
1996
8*prime solutions:
1016
1048
1096
1112
1192
1208
1256
1304
1336
1384
1432
1448
1528
1544
1576
1592
1688
1784
1816
1832
1864
1912
1928
16*prime solutions:
1072
1136
1168
1264
1328
1424
1552
1616
1648
1712
1744
1808
Jimmygu3
Feb 26, 2014

I believe this answer, which has stood unchallenged for over 10 years, is incorrect.

Because the consecutive numbers must be nonnegative, a double prime like 1006 won't work. As javaguru stated, we would need to center 503 consecutive digits on 2. That string would begin with -249, which is not allowed. Extending javaguru's analysis, we see that x-1≤2y, where x is the lowest odd factor. Any power of 2 (y) multiplied by a prime (x) that is greater than 2y+1, is a solution.

If you think I'm wrong, let me know! I'm seeing 150 additional solutions.

2*prime solutions:
1006
1018
1042
1046
1082
1094
1114
1126
1138
1142
1154
1174
1186
1198
1202
1214
1226
1234
1238
1262
1282
1286
1294
1306
1318
1322
1346
1354
1366
1382
1402
1418
1438
1454
1466
1478
1486
1502
1514
1522
1538
1546
1574
1594
1618
1622
1642
1646
1654
1658
1678
1706
1714
1718
1726
1754
1762
1766
1774
1814
1822
1838
1858
1874
1882
1894
1906
1934
1942
1954
1966
1982
1994
4*prime solutions:
1004
1028
1052
1076
1084
1108
1124
1132
1172
1228
1244
1252
1268
1324
1348
1388
1396
1412
1436
1468
1492
1516
1532
1556
1588
1604
1636
1676
1684
1724
1732
1756
1772
1796
1828
1844
1852
1868
1916
1948
1964
1996
8*prime solutions:
1016
1048
1096
1112
1192
1208
1256
1304
1336
1384
1432
1448
1528
1544
1576
1592
1688
1784
1816
1832
1864
1912
1928
16*prime solutions:
1072
1136
1168
1264
1328
1424
1552
1616
1648
1712
1744
1808
Jimmygu3
Feb 26, 2014

Sorry for the double post. The first was posted with an error: x-1≤y instead of x-1≤2y.
Gerd*de*
Feb 26, 2014

Dear Jimmy, you've done a lot of work, but I'm sorry - in fact it is wrong.

I hope it's enough to show the first 2 of your numbers.

1006 -> 250 + 251 + 252 + 253
1018 -> 253 + 254 + 255 + 256

Hae a good time!
Gerd
Gerd*de*
Feb 26, 2014

Of course it's 'Have a good time'.
The 'V' on my keyboard has gone making holidays.
Jimmygu3
Feb 26, 2014

Thanks, Gerd! I should have known that wouldn't have gone unnoticed for 10 years! Wish I could delete the posts (or at least one of 'em), but admin says it can't be done. I'll be more careful before I post next time!



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