### Brain Teasers

# Consecutive numbers

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?

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

There is only one number that you cannot get.

What is this number?

### Hint

I hope you got the power!### Answer

1024Only powers of 2 are not reachable, and the next number is 2048.

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## Comments

Great teaser. Any multiple of an odd number is reachable and that leaves only the powers of two.

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

Thank you, very well done, a great teaching tool.

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.

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.

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-1â‰¤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

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-1â‰¤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

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-1â‰¤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

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-1â‰¤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

Sorry for the double post. The first was posted with an error: x-1â‰¤y instead of x-1â‰¤2y.

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

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

Of course it's 'Have a good time'.

The 'V' on my keyboard has gone making holidays.

The 'V' on my keyboard has gone making holidays.

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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