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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

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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 krishnan 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 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 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 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!