Open Middle®
Directions: Using the digits 1-9 at most one time each, find the closest sum to 10,000 using two 4-digit addends.
Hint
Which digits will be the most helpful to focus on to make the solution as close to 10,000 as possible?
Answer
10,000 = (6521+3479) or (5671+4329) or (5679+4321) & other mixes of the same numbers in different positions
Source: Jessica Goree
7351+2649=10,000
7,351+2,649=10,000
All 48 solutions (excluding the ones where the addends are swapped):
2 341 + 7 659 = 10 000
2 349 + 7 651 = 10 000
2 351 + 7 649 = 10 000
2 359 + 7 641 = 10 000
2 431 + 7 569 = 10 000
2 439 + 7 561 = 10 000
2 461 + 7 539 = 10 000
2 469 + 7 531 = 10 000
2 531 + 7 469 = 10 000
2 539 + 7 461 = 10 000
2 561 + 7 439 = 10 000
2 569 + 7 431 = 10 000
2 641 + 7 359 = 10 000
2 649 + 7 351 = 10 000
2 651 + 7 349 = 10 000
2 659 + 7 341 = 10 000
3 241 + 6 759 = 10 000
3 249 + 6 751 = 10 000
3 251 + 6 749 = 10 000
3 259 + 6 741 = 10 000
3 421 + 6 579 = 10 000
3 429 + 6 571 = 10 000
3 471 + 6 529 = 10 000
3 479 + 6 521 = 10 000
3 521 + 6 479 = 10 000
3 529 + 6 471 = 10 000
3 571 + 6 429 = 10 000
3 579 + 6 421 = 10 000
3 741 + 6 259 = 10 000
3 749 + 6 251 = 10 000
3 751 + 6 249 = 10 000
3 759 + 6 241 = 10 000
4 231 + 5 769 = 10 000
4 239 + 5 761 = 10 000
4 261 + 5 739 = 10 000
4 269 + 5 731 = 10 000
4 321 + 5 679 = 10 000
4 329 + 5 671 = 10 000
4 371 + 5 629 = 10 000
4 379 + 5 621 = 10 000
4 621 + 5 379 = 10 000
4 629 + 5 371 = 10 000
4 671 + 5 329 = 10 000
4 679 + 5 321 = 10 000
4 731 + 5 269 = 10 000
4 739 + 5 261 = 10 000
4 761 + 5 239 = 10 000
4 769 + 5 231 = 10 000
All solutions have a 1 and 9 in the ones position.
The sums closest to – but not equal to 10 000 – are 9 999 (192 solutions excluding ones where the addends are swapped) and 10 001 (48 solutions excluding ones where the addends are swapped)
1234 + 8765 = 9999
1235 + 8764 = 9999
1243 + 8756 = 9999
1246 + 8753 = 9999
1253 + 8746 = 9999
1256 + 8743 = 9999
1264 + 8735 = 9999
1265 + 8734 = 9999
1324 + 8675 = 9999
1325 + 8674 = 9999
1342 + 8657 = 9999
1347 + 8652 = 9999
1352 + 8647 = 9999
1357 + 8642 = 9999
1374 + 8625 = 9999
1375 + 8624 = 9999
1423 + 8576 = 9999
1426 + 8573 = 9999
1432 + 8567 = 9999
1437 + 8562 = 9999
1462 + 8537 = 9999
1467 + 8532 = 9999
1473 + 8526 = 9999
1476 + 8523 = 9999
1523 + 8476 = 9999
1526 + 8473 = 9999
1532 + 8467 = 9999
1537 + 8462 = 9999
1562 + 8437 = 9999
1567 + 8432 = 9999
1573 + 8426 = 9999
1576 + 8423 = 9999
1624 + 8375 = 9999
1625 + 8374 = 9999
1642 + 8357 = 9999
1647 + 8352 = 9999
1652 + 8347 = 9999
1657 + 8342 = 9999
1674 + 8325 = 9999
1675 + 8324 = 9999
1734 + 8265 = 9999
1735 + 8264 = 9999
1743 + 8256 = 9999
1746 + 8253 = 9999
1753 + 8246 = 9999
1756 + 8243 = 9999
1764 + 8235 = 9999
1765 + 8234 = 9999
2134 + 7865 = 9999
2135 + 7864 = 9999
2143 + 7856 = 9999
2146 + 7853 = 9999
2153 + 7846 = 9999
2156 + 7843 = 9999
2164 + 7835 = 9999
2165 + 7834 = 9999
2314 + 7685 = 9999
2315 + 7684 = 9999
2341 + 7658 = 9999
2348 + 7651 = 9999
2351 + 7648 = 9999
2358 + 7641 = 9999
2384 + 7615 = 9999
2385 + 7614 = 9999
2413 + 7586 = 9999
2416 + 7583 = 9999
2431 + 7568 = 9999
2438 + 7561 = 9999
2461 + 7538 = 9999
2468 + 7531 = 9999
2483 + 7516 = 9999
2486 + 7513 = 9999
2513 + 7486 = 9999
2516 + 7483 = 9999
2531 + 7468 = 9999
2538 + 7461 = 9999
2561 + 7438 = 9999
2568 + 7431 = 9999
2583 + 7416 = 9999
2586 + 7413 = 9999
2614 + 7385 = 9999
2615 + 7384 = 9999
2641 + 7358 = 9999
2648 + 7351 = 9999
2651 + 7348 = 9999
2658 + 7341 = 9999
2684 + 7315 = 9999
2685 + 7314 = 9999
2834 + 7165 = 9999
2835 + 7164 = 9999
2843 + 7156 = 9999
2846 + 7153 = 9999
2853 + 7146 = 9999
2856 + 7143 = 9999
2864 + 7135 = 9999
2865 + 7134 = 9999
3124 + 6875 = 9999
3125 + 6874 = 9999
3142 + 6857 = 9999
3147 + 6852 = 9999
3152 + 6847 = 9999
3157 + 6842 = 9999
3174 + 6825 = 9999
3175 + 6824 = 9999
3214 + 6785 = 9999
3215 + 6784 = 9999
3241 + 6758 = 9999
3248 + 6751 = 9999
3251 + 6748 = 9999
3258 + 6741 = 9999
3284 + 6715 = 9999
3285 + 6714 = 9999
3412 + 6587 = 9999
3417 + 6582 = 9999
3421 + 6578 = 9999
3428 + 6571 = 9999
3471 + 6528 = 9999
3478 + 6521 = 9999
3482 + 6517 = 9999
3487 + 6512 = 9999
3512 + 6487 = 9999
3517 + 6482 = 9999
3521 + 6478 = 9999
3528 + 6471 = 9999
3571 + 6428 = 9999
3578 + 6421 = 9999
3582 + 6417 = 9999
3587 + 6412 = 9999
3714 + 6285 = 9999
3715 + 6284 = 9999
3741 + 6258 = 9999
3748 + 6251 = 9999
3751 + 6248 = 9999
3758 + 6241 = 9999
3784 + 6215 = 9999
3785 + 6214 = 9999
3824 + 6175 = 9999
3825 + 6174 = 9999
3842 + 6157 = 9999
3847 + 6152 = 9999
3852 + 6147 = 9999
3857 + 6142 = 9999
3874 + 6125 = 9999
3875 + 6124 = 9999
4123 + 5876 = 9999
4126 + 5873 = 9999
4132 + 5867 = 9999
4137 + 5862 = 9999
4162 + 5837 = 9999
4167 + 5832 = 9999
4173 + 5826 = 9999
4176 + 5823 = 9999
4213 + 5786 = 9999
4216 + 5783 = 9999
4231 + 5768 = 9999
4238 + 5761 = 9999
4261 + 5738 = 9999
4268 + 5731 = 9999
4283 + 5716 = 9999
4286 + 5713 = 9999
4312 + 5687 = 9999
4317 + 5682 = 9999
4321 + 5678 = 9999
4328 + 5671 = 9999
4371 + 5628 = 9999
4378 + 5621 = 9999
4382 + 5617 = 9999
4387 + 5612 = 9999
4612 + 5387 = 9999
4617 + 5382 = 9999
4621 + 5378 = 9999
4628 + 5371 = 9999
4671 + 5328 = 9999
4678 + 5321 = 9999
4682 + 5317 = 9999
4687 + 5312 = 9999
4713 + 5286 = 9999
4716 + 5283 = 9999
4731 + 5268 = 9999
4738 + 5261 = 9999
4761 + 5238 = 9999
4768 + 5231 = 9999
4783 + 5216 = 9999
4786 + 5213 = 9999
4823 + 5176 = 9999
4826 + 5173 = 9999
4832 + 5167 = 9999
4837 + 5162 = 9999
4862 + 5137 = 9999
4867 + 5132 = 9999
4873 + 5126 = 9999
4876 + 5123 = 9999
1342 + 8659 = 10001
1349 + 8652 = 10001
1352 + 8649 = 10001
1359 + 8642 = 10001
1432 + 8569 = 10001
1439 + 8562 = 10001
1462 + 8539 = 10001
1469 + 8532 = 10001
1532 + 8469 = 10001
1539 + 8462 = 10001
1562 + 8439 = 10001
1569 + 8432 = 10001
1642 + 8359 = 10001
1649 + 8352 = 10001
1652 + 8349 = 10001
1659 + 8342 = 10001
3142 + 6859 = 10001
3149 + 6852 = 10001
3152 + 6849 = 10001
3159 + 6842 = 10001
3412 + 6589 = 10001
3419 + 6582 = 10001
3482 + 6519 = 10001
3489 + 6512 = 10001
3512 + 6489 = 10001
3519 + 6482 = 10001
3582 + 6419 = 10001
3589 + 6412 = 10001
3842 + 6159 = 10001
3849 + 6152 = 10001
3852 + 6149 = 10001
3859 + 6142 = 10001
4132 + 5869 = 10001
4139 + 5862 = 10001
4162 + 5839 = 10001
4169 + 5832 = 10001
4312 + 5689 = 10001
4319 + 5682 = 10001
4382 + 5619 = 10001
4389 + 5612 = 10001
4612 + 5389 = 10001
4619 + 5382 = 10001
4682 + 5319 = 10001
4689 + 5312 = 10001
4832 + 5169 = 10001
4839 + 5162 = 10001
4862 + 5139 = 10001
4869 + 5132 = 10001
9,000+1,000=10,000
8,000+2,000=10,000