## Poll

10 votes (45.45%) | |||

9 votes (40.9%) | |||

5 votes (22.72%) | |||

2 votes (9.09%) | |||

7 votes (31.81%) | |||

3 votes (13.63%) | |||

4 votes (18.18%) | |||

3 votes (13.63%) | |||

9 votes (40.9%) | |||

6 votes (27.27%) |

**22 members have voted**

Quote:teliotOn the topic of Golfing, let me create a different problem that is based on efficient stamp totals.

Suppose the Post Office issues two values of stamps X cents and Y cents. What are the most efficient values for these two stamps X & Y so that we can make each postage from $1.20 to $2.00, using the least number of stamps overall?

Yeah, "Easy" Math Puzzles. :)

My quick guess is 12 cents and 11 cents.

We need two numbers -one, call it n, that is even and the other that has multiples that are capable of making xmodn, x=1...n-1.

Your answer requires a total of 1118 stamps. My answer requires 1072 stamps.Quote:gordonm888

My quick guess is 12 cents and 11 cents.

We need two numbers -one, call it n, that is even and the other that has multiples that are capable of making xmodn, x=1...n-1.

Quote:teliot...My answer requires 1072 stamps.

Yes!Quote:charliepatrickI think you accidentally (or not) gave a hint earlier as 120 = (13-1)*(11-1) and both 13 and 11 are primes; and they seem to work. The early values can be achieved in either 10 or 11 stamps, e.g 120 = 5*each, 121=11*11c, 122 is 4*11c+6*13c, so to get 2 more you replace one 11 with a 13. At the end 199 needs 17 stamps, and 200 (which is 208-8 so 12*13+4*11 requires 16 stamps.) The total is 1072.

Now, what's the answer if we also need to make a postage of $1.19? This is where it gets slightly evil.

1, 2, Total = 6560

1, 3, Total = 4415

1, 4, Total = 3362

1, 5, Total = 2747

1, 6, Total = 2349

1, 7, Total = 2075

1, 8, Total = 1886

1, 9, Total = 1743

1, 10, Total = 1640

1, 11, Total = 1559

1, 12, Total = 1496

1, 13, Total = 1451

1, 14, Total = 1431

1, 15, Total = 1389

1, 16, Total = 1394

1, 17, Total = 1367

1, 18, Total = 1400

1, 19, Total = 1379

1, 20, Total = 1394

1, 21, Total = 1399

1, 22, Total = 1445

1, 23, Total = 1419

1, 24, Total = 1395

1, 25, Total = 1511

1, 26, Total = 1504

1, 27, Total = 1509

1, 28, Total = 1550

1, 29, Total = 1571

1, 30, Total = 1508

1, 31, Total = 1559

1, 32, Total = 1640

1, 33, Total = 1751

1, 34, Total = 1793

1, 35, Total = 1757

1, 36, Total = 1739

1, 37, Total = 1739

1, 38, Total = 1757

1, 39, Total = 1793

1, 40, Total = 1886

1, 41, Total = 1919

1, 42, Total = 1927

1, 43, Total = 1949

1, 44, Total = 1985

1, 45, Total = 2035

1, 46, Total = 2099

1, 47, Total = 2177

1, 48, Total = 2269

1, 49, Total = 2375

1, 50, Total = 2495

1, 51, Total = 2479

1, 52, Total = 2420

1, 53, Total = 2367

1, 54, Total = 2320

1, 55, Total = 2279

1, 56, Total = 2244

1, 57, Total = 2215

1, 58, Total = 2192

1, 59, Total = 2175

1, 60, Total = 2223

1, 61, Total = 2339

1, 62, Total = 2465

1, 63, Total = 2601

1, 64, Total = 2747

1, 65, Total = 2903

1, 66, Total = 3069

1, 67, Total = 3245

1, 68, Total = 3230

1, 69, Total = 3219

1, 70, Total = 3212

1, 71, Total = 3209

1, 72, Total = 3210

1, 73, Total = 3215

1, 74, Total = 3224

1, 75, Total = 3237

1, 76, Total = 3254

1, 77, Total = 3275

1, 78, Total = 3300

1, 79, Total = 3329

1, 80, Total = 3362

1, 81, Total = 3399

1, 82, Total = 3440

1, 83, Total = 3485

1, 84, Total = 3534

1, 85, Total = 3587

1, 86, Total = 3644

1, 87, Total = 3705

1, 88, Total = 3770

1, 89, Total = 3839

1, 90, Total = 3912

1, 91, Total = 3989

1, 92, Total = 4070

1, 93, Total = 4155

1, 94, Total = 4244

1, 95, Total = 4337

1, 96, Total = 4434

1, 97, Total = 4535

1, 98, Total = 4640

1, 99, Total = 4749

1, 100, Total = 4862

1, 101, Total = 4879

1, 102, Total = 4797

1, 103, Total = 4715

1, 104, Total = 4633

1, 105, Total = 4551

1, 106, Total = 4469

1, 107, Total = 4387

1, 108, Total = 4305

1, 109, Total = 4223

1, 110, Total = 4141

1, 111, Total = 4059

1, 112, Total = 3977

1, 113, Total = 3895

1, 114, Total = 3813

1, 115, Total = 3731

1, 116, Total = 3649

1, 117, Total = 3567

1, 118, Total = 3485

1, 119, Total = 3403

2, 3, Total = 4387

2, 5, Total = 2713

2, 7, Total = 2042

2, 9, Total = 1706

2, 11, Total = 1522

2, 13, Total = 1419

2, 15, Total = 1359

2, 17, Total = 1342

2, 19, Total = 1346

2, 21, Total = 1362

2, 23, Total = 1384

2, 25, Total = 1445

2, 27, Total = 1477

2, 29, Total = 1531

2, 31, Total = 1537

2, 33, Total = 1657

2, 35, Total = 1705

2, 37, Total = 1727

2, 39, Total = 1785

2, 41, Total = 1879

2, 43, Total = 1927

2, 45, Total = 2003

2, 47, Total = 2107

2, 49, Total = 2239

2, 51, Total = 2350

2, 53, Total = 2332

2, 55, Total = 2326

2, 57, Total = 2332

2, 59, Total = 2350

2, 61, Total = 2439

2, 63, Total = 2605

2, 65, Total = 2791

2, 67, Total = 2997

2, 69, Total = 3022

2, 71, Total = 3055

2, 73, Total = 3096

2, 75, Total = 3145

2, 77, Total = 3202

2, 79, Total = 3267

2, 81, Total = 3340

2, 83, Total = 3421

2, 85, Total = 3510

2, 87, Total = 3607

2, 89, Total = 3712

2, 91, Total = 3825

2, 93, Total = 3946

2, 95, Total = 4075

2, 97, Total = 4212

2, 99, Total = 4357

2, 101, Total = 4510

2, 103, Total = 4469

2, 105, Total = 4428

2, 107, Total = 4387

2, 109, Total = 4346

2, 111, Total = 4305

2, 113, Total = 4264

2, 115, Total = 4223

2, 117, Total = 4182

2, 119, Total = 4141

3, 4, Total = 3300

3, 5, Total = 2681

3, 7, Total = 2009

3, 8, Total = 1813

3, 10, Total = 1562

3, 11, Total = 1485

3, 13, Total = 1383

3, 14, Total = 1353

3, 16, Total = 1322

3, 17, Total = 1303

3, 19, Total = 1309

3, 20, Total = 1311

3, 22, Total = 1345

3, 23, Total = 1353

3, 25, Total = 1397

3, 26, Total = 1408

3, 28, Total = 1468

3, 29, Total = 1491

3, 31, Total = 1513

3, 32, Total = 1566

3, 34, Total = 1642

3, 35, Total = 1661

3, 37, Total = 1685

3, 38, Total = 1723

3, 40, Total = 1782

3, 41, Total = 1839

3, 43, Total = 1893

3, 44, Total = 1927

3, 46, Total = 2009

3, 47, Total = 2057

3, 49, Total = 2167

3, 50, Total = 2229

3, 52, Total = 2269

3, 53, Total = 2293

3, 55, Total = 2297

3, 56, Total = 2328

3, 58, Total = 2343

3, 59, Total = 2381

4, 5, Total = 2648

4, 7, Total = 1973

4, 9, Total = 1636

4, 11, Total = 1448

4, 13, Total = 1346

4, 15, Total = 1287

4, 17, Total = 1271

4, 19, Total = 1271

4, 21, Total = 1285

4, 23, Total = 1308

4, 25, Total = 1343

4, 27, Total = 1401

4, 29, Total = 1451

4, 31, Total = 1481

4, 33, Total = 1566

4, 35, Total = 1619

4, 37, Total = 1661

4, 39, Total = 1721

5, 6, Total = 2214

5, 7, Total = 1939

5, 8, Total = 1741

5, 9, Total = 1599

5, 11, Total = 1411

5, 12, Total = 1353

5, 13, Total = 1307

5, 14, Total = 1282

5, 16, Total = 1243

5, 17, Total = 1231

5, 18, Total = 1230

5, 19, Total = 1227

5, 21, Total = 1243

5, 22, Total = 1266

5, 23, Total = 1273

5, 24, Total = 1282

5, 26, Total = 1339

5, 27, Total = 1353

5, 28, Total = 1383

5, 29, Total = 1411

6, 7, Total = 1904

6, 11, Total = 1374

6, 13, Total = 1271

6, 17, Total = 1188

6, 19, Total = 1194

6, 23, Total = 1242

7, 8, Total = 1670

7, 9, Total = 1527

7, 10, Total = 1418

7, 11, Total = 1337

7, 12, Total = 1277

7, 13, Total = 1235

7, 15, Total = 1177

7, 16, Total = 1160

7, 17, Total = 1157

7, 18, Total = 1155

7, 19, Total = 1157

7, 20, Total = 1159

8, 9, Total = 1490

8, 11, Total = 1300

8, 13, Total = 1198

8, 15, Total = 1137

8, 17, Total = 1123

9, 10, Total = 1344

9, 11, Total = 1263

9, 13, Total = 1159

9, 14, Total = 1121

10, 11, Total = 1226

10, 13, Total = 1121

11, 12, Total = 1128

Quote:teliotOn the topic of Golfing, let me create a different problem that is based on efficient stamp totals.

Suppose the Post Office issues two values of stamps X cents and Y cents. What are the most efficient values for these two stamps X & Y so that we can make each postage from $1.20 to $2.00, using the least number of stamps overall?

Yeah, "Easy" Math Puzzles. :)

Because I have nothing better to do at 7 AM on a Saturday morning...

(13, 11) uses 1072 stamps:

Total | 13 | 11 | Total | 13 | 11 | Total | 13 | 11 | Total | 13 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

120 | 5 | 5 | 140 | 4 | 8 | 160 | 3 | 11 | 180 | 13 | 1 |

121 | 0 | 11 | 141 | 10 | 1 | 161 | 9 | 4 | 181 | 8 | 7 |

122 | 6 | 4 | 142 | 5 | 7 | 162 | 4 | 10 | 182 | 14 | 0 |

123 | 1 | 10 | 143 | 11 | 0 | 163 | 10 | 3 | 183 | 9 | 6 |

124 | 7 | 3 | 144 | 6 | 6 | 164 | 5 | 9 | 184 | 4 | 12 |

125 | 2 | 9 | 145 | 1 | 12 | 165 | 11 | 2 | 185 | 10 | 5 |

126 | 8 | 2 | 146 | 7 | 5 | 166 | 6 | 8 | 186 | 5 | 11 |

127 | 3 | 8 | 147 | 2 | 11 | 167 | 12 | 1 | 187 | 11 | 4 |

128 | 9 | 1 | 148 | 8 | 4 | 168 | 7 | 7 | 188 | 6 | 10 |

129 | 4 | 7 | 149 | 3 | 10 | 169 | 13 | 0 | 189 | 12 | 3 |

130 | 10 | 0 | 150 | 9 | 3 | 170 | 8 | 6 | 190 | 7 | 9 |

131 | 5 | 6 | 151 | 4 | 9 | 171 | 3 | 12 | 191 | 13 | 2 |

132 | 0 | 12 | 152 | 10 | 2 | 172 | 9 | 5 | 192 | 8 | 8 |

133 | 6 | 5 | 153 | 5 | 8 | 173 | 4 | 11 | 193 | 14 | 1 |

134 | 1 | 11 | 154 | 11 | 1 | 174 | 10 | 4 | 194 | 9 | 7 |

135 | 7 | 4 | 155 | 6 | 7 | 175 | 5 | 10 | 195 | 15 | 0 |

136 | 2 | 10 | 156 | 12 | 0 | 176 | 11 | 3 | 196 | 10 | 6 |

137 | 8 | 3 | 157 | 7 | 6 | 177 | 6 | 9 | 197 | 5 | 12 |

138 | 3 | 9 | 158 | 2 | 12 | 178 | 12 | 2 | 198 | 11 | 5 |

139 | 9 | 2 | 159 | 8 | 5 | 179 | 7 | 8 | 199 | 6 | 11 |

200 | 12 | 4 |

Ha! I was writing c code at 5:30 this morning.Quote:ThatDonGuyBecause I have nothing better to do at 7 AM on a Saturday morning...

(13, 11) uses 1072 stamps:

Total 13 11 Total 13 11 Total 13 11 Total 13 11 120 5 5 140 4 8 160 3 11 180 13 1 121 0 11 141 10 1 161 9 4 181 8 7 122 6 4 142 5 7 162 4 10 182 14 0 123 1 10 143 11 0 163 10 3 183 9 6 124 7 3 144 6 6 164 5 9 184 4 12 125 2 9 145 1 12 165 11 2 185 10 5 126 8 2 146 7 5 166 6 8 186 5 11 127 3 8 147 2 11 167 12 1 187 11 4 128 9 1 148 8 4 168 7 7 188 6 10 129 4 7 149 3 10 169 13 0 189 12 3 130 10 0 150 9 3 170 8 6 190 7 9 131 5 6 151 4 9 171 3 12 191 13 2 132 0 12 152 10 2 172 9 5 192 8 8 133 6 5 153 5 8 173 4 11 193 14 1 134 1 11 154 11 1 174 10 4 194 9 7 135 7 4 155 6 7 175 5 10 195 15 0 136 2 10 156 12 0 176 11 3 196 10 6 137 8 3 157 7 6 177 6 9 197 5 12 138 3 9 158 2 12 178 12 2 198 11 5 139 9 2 159 8 5 179 7 8 199 6 11 200 12 4

Correct, of course.

Yes, the coinages must be relatively prime. If d is a divisor of a & b, then d is also a divisor of ax + by for all x, y. Ergo, you can only make coinages that are divisible by gcd(a,b).Quote:WizardI'm sure we're all wondering if there is at least a short cut to the right answer to such problems or is brute force the only way? I suspect in general the coinages will be semi-prime to each other, but I can't put into words why.

As for your interesting question of getting a short cut, that's why I gave the example of $1.19. The answer doesn't fit the most obvious short cut, namely to minimize the difference |a-b| while maximizing the product (a-1)*(b-1) <= N (where N is the smallest coinage that needs to be made).

For instance if you were using 5 and 7, 25 would be 5*5 and 0*7. Reducing the number of 5's by 1 and adding a 7 instead, would add 2 (27=4*5+1*7 etc.). So this creates 25 27 29 31 33 35. 28 is 4*7 so 26 is 3*7+1*5. 30 is 6*5 which starts the next series up to 42; 7*5 starts 35....

Presumably this process continues all the way up. So you might be able to determine how this goes based on the pattern and end conditions.