stands for some digit, there is only one solution to the equation + = that uses the same digit every time, namely 0 + 0 = 0. Similarly, the only solution to = that uses exactly two different digits is 52 = 25.For a given set S of n digits, what is the digit equation using the fewest digits with only one solution using n different digits, and the solution uses the digits from S?
When n=1, a computer search quickly finds all of the smallest solutions below. But for n=2 and n=3, there are many unsolved cases. Can you find or improve the solutions below? What are the shortest solutions for n=10, n=9, and n=8?
Here are the shortest-known solutions:
| 0 | 0 + 0 = 0 |
| 1 | 1 / 1 = 1 11 = 1 |
| 2 | 22 = 2 + 2 (2 + 2) / 2 = 2 |
| 3 | (3 + 3 + 3) / 3 = 3 |
| 4 | (4 + 4 + 4 + 4) / 4 = 4 |
| 5 | 5 × (5 + 5) + 5 = 55 |
| 6 | 6 × (6 + 6) = 66 + 6 |
| 7 | 7 × 7 + 7 + 7 + 7 + 7 = 77 |
| 8 | 8 × 8 + 8 + 8 + 8 = 88 |
| 9 | 9 × 9 + 9 + 9 = 99 |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 1 | 1001+1 = 10000 | ||||||||
| 2 | (2 × 2)2 + 22 = 20 22×2 + 22 = 20 | 112 = 121 | |||||||
| 3 | 33 = 30 – 3 / 30 (3 / 30)3 = 30 – 3 33+0/3 = 30 – 3 33–0/3 = 30 – 3 33–0–0 = 30 – 3 33–03 = 30 – 3 33/30 = 30 – 3 33×30 = 30 – 3 3330 = 30 – 3 (33 + 3) / 30 = 30 (33 + 3)30 = 30 | 113 = 1331 33 – 1 = (3 – 1) × 13 | 2 × 222 = 32 2 × 22+2 = 32 2 × 22×2 = 32 2 × (2 + 2)2 = 32 2 × (2 × 2)2 = 32 (2 × 2)3 / 2 = 32 23+3 / 2 = 32 2 × 2 × 23 = 32 33 – 22 = 23 | ||||||
| 4 | 4 × (40 / 4)4 = 40000 | 11 + 1 = 4 + 4 + 4 / 1 11 + 1 = 4 + (4 + 4) / 1 11 + 1 = (4 + 4 + 4) / 1 11 + 1 × 1 = 4 + 4 + 4 11 = 4 + 4 + (4 – 1) / 1 | 22 + 24 / 2 = 4 × 4 | 344 / 43 = 43 / (4 + 4) | |||||
| 5 | 5 / 50 + 5 = 50 / 5 50 × 50 / 5 = 5 + 5 (5 + 5)50 = 50 / 5 (5 × (5 + 5) )50 = 50 | 5 – 1 – 1 = 15 / 5 | 52 = 25 | (53 – 5) / 3 = 35 + 5 | (5 – 4) × (44 + 44 + 4) = 555 + 5 – 44 (Berend van der Zwaag) | ||||
| 6 | 6 × (60 / 6)6 = 6000000 | 6 + 6 = 11 + 1 | 26 = 62 + 2 | 6 × 6 = 33 + 3 66/3 = 36 | 4 × 4 × 4 = 64 (6 – 4)6 = 64 | (5 × 65 + 6 + 6) / (6 + 6 – 5) = 5556 | |||
| 7 | 7 × (70 / 7)7 = 70000000 | 11 + 1 + 1 + 1 = 7 + 7 | (27 + 7) / (7 – 2) = 27 | 33 + 37 = (7 – 3)3 | ? | 57 – 55 / 5 + 55 × 5 = 77775 | 66 / 6 = 7776 | ||
| 8 | 8 × (80 / 8)8 = 800000000 | 11 + 1 – 8 = 8 / (1 + 1) | (8 × 8 – 8) / 2 = 28 (82 – 8) / 2 = 28 | 38–3 × (8+8) = 3888 (Bryce Herdt) | 8 × 8 – 4 × 4 = 48 | ? | 8 × 86 = 688 | ? | |
| 9 | 9 × (9 – 0) – 0 = 90 – 9 9 × 9 = 90 × (90 – 9) 9 × 990 = 90 – 9 9 × 9 = 90 – 9 / 90 (9 × 9)90 = 90 – 9 9 + 9 / 9 = 90 / (9 – 0) 9 + 90 = 90 / (9 – 0) 9 × (9 + 0 + 0) = 90 – 9 (9 × 9 + 9)90 = 90 | (111 – 99 – 9)11–9 = 9 | (92 + 92)2 = 29929 (Bryce Herdt) | ? | ? | 5 × 5 × (5 × 5 + 9 + 9 – 5) = 59999 – 95 | ? | (9 – 7)7 = 79 + 7 × 7 | 9 × 9 / (9 – 8) = (89 – 8) 9 × 99–8 = 89 – 8 (9 × 9)9–8 = 89 – 8 (9 × 9 + 8)9–8 = 89 |
| 0+ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 2 | 1010/2 = 100000 (Andrew Bayly) | |||||||
| 3 | 1033–3×3–3–3 = 1000000000000 (Andrew Bayly) | ? | ||||||
| 4 | 1004 = 100000000 (Andrew Bayly) | ? | ? | |||||
| 5 | 105 = 100000 (5 + 5)5 = 100000 | ? | ? | ? | ||||
| 6 | 106 = 1000000 | ? | ? | (46–4 + 4)6 = 64000000 (Andrew Bayly) | ? | |||
| 7 | 107 = 10000000 | ? | ? | ? | ? | 606+6–7 = 777600000 (Andrew Bayly) | ||
| 8 | 108 = 100000000 | ? | ? | ? | ? | ? | ? | |
| 9 | 109 = 1000000000 | ? | ? | ? | ? | ? | ? | ? |
| 1+ | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | |||||||
| 4 | ? | ? | |||||
| 5 | (5 / 1 / 1)2 = 25 | ? | ? | ||||
| 6 | 66/2 = 216 | ? | 66 / 4 = 11664 (Andrew Bayly) | 166 – 115 = 16616165 (Andrew Bayly) | |||
| 7 | ? | ? | (1 + 1 + 1)11 = 177147 (Andrew Bayly) | ? | (7 – 1)6–1 = 7776 (Andrew Bayly) | ||
| 8 | 8 × (8 + 8) = 128 | (33 – 1)8–3 = 11881188 + 188 (Andrew Bayly) | (8 – 1 – 1 – 1)11 = 48818841 + 844 × 11 (Andrew Bayly) | ? | ? | ? | |
| 9 | ( (2 × (9 – 2 × 2) )9 – 1) / (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) = 111111111 (Andrew Bayly) | ? | ? | ? | ? | ? | ? |
| 2+ | 3 | 4 | 5 | 6 | 7 | 8 |
| 4 | 224 – 322 = 233232 (Andrew Bayly) | |||||
| 5 | ? | ? | ||||
| 6 | ? | ? | 52×2 = 625 522 = 625 | |||
| 7 | ? | ? | ? | ? | ||
| 8 | (82 – 2)3 = 238328 (Andrew Bayly) | ? | ? | ? | ? | |
| 9 | ? | (94 – 4)2 = 42994249 (Andrew Bayly) | ? | ? | ? | ? |
| 3+ | 4 | 5 | 6 | 7 | 8 |
| 5 | ? | ||||
| 6 | (33 – 4)3×3–4 = 6436343 (Andrew Bayly) | ? | |||
| 7 | 73 = 343 | ? | 67 = 36 × 7776 | ||
| 8 | ? | ? | ? | ? | |
| 9 | ? | ? | ? | ? | ? |
| 4+ | 5 | 6 | 7 | 8 |
| 6 | 6 × 65 = 46656 | |||
| 7 | ? | 6 × 64 = 7776 | ||
| 8 | ? | 66 = 46648 + 8 | ? | |
| 9 | ? | ? | ? | ? |
| 5+ | 6 | 7 | 8 |
| 7 | 65 = 7776 | ||
| 8 | ? | ? | |
| 9 | ? | ? | ? |
| 6+ | 7 | 8 |
| 8 | ? | |
| 9 | ? | ? |
| 7+ | 8 |
| 9 | ? |
| all but | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 1 | (9 – 7)(8/4)3 = 256 (8 / 4)(9–7)3 = 256 | ||||||||
| 2 | ? | ? | |||||||
| 3 | (8 / 4)17–9 = 256 2418/9 = 576 | ? | ? | ||||||
| 4 | (35 / 81)6 = 729 | ? | 381 + 9 = 6570 | 6(5–0)/1 / 8 = 972 65/1–0 / 8 = 972 651–0 / 8 = 972 6(5–0)1 / 8 = 972 610–5 / 8 = 972 | |||||
| 5 | (78 – 43)2 = 196 | (27 / 8)3 = 4096 | 8(7–3)/1 = 4096 87/1–3 = 4096 | 786/9 = 2401 | ? | ||||
| 6 | 8(7–4)3/9 = 512 (83 / 4)9/7 = 512 | ? | 3151 = 14348907 (Andrew Bayly) | (8 / (9 – 7) )5 = 1024 5 × 47 = 81920 | ? | (9 – 0)4 / 3 = 2187 | |||
| 7 | 95×8/16 = 243 834/9–6 = 512 (46/3 / 8)9 = 512 ( (6 / 3)4 / 8)9 = 512 | ? | ? | 215 / 8 = 4096 | 893/6–0 = 512 8(9–0)3/6 = 512 | ? | 95/(10–8) = 243 | ||
| 8 | (54 / 6)31 = 729 (54 / 61)3 = 729 (541 / 6)3 = 729 361×(5–4) = 729 3(6×(5–4))1 = 729 | ? | ? | (9 – 7)10 / 4 = 256 | 7222 = 33232930569601 (Andrew Bayly) | 1210 = 61917364224 (Andrew Bayly) | (7 – 9 / 3)5 = 1024 | 362 = 150094635296999121 (Andrew Bayly) | |
| 9 | 867–43 = 512 | ? | ? | 2410–8 = 576 8421 = 7056 (84 × 1)2 = 7056 | ? | 3 × 210 / 4 = 768 | 78/(5–3) = 2401 | 65 / 3 / 24 = 108 803 / 46 = 125 | ? |
| all but 0 | 694/2 / 3 = 1587 (78/4 – 5)2 = 1936 (72 – 5)8/4 = 1936 354/6 / 9 = 2187 (9 / (6 – 5) )4 / 3 = 2187 94/(6–5) / 3 = 2187 5927–6 = 3481 723×6/9 = 5184 (83 – 9)21 = 5476 98/2 + 13 = 6574 (3 × 76)2 = 51984 |
| all but 1 | 853–72 = 4096 823/(7–5) = 4096 (84 × 3)2 / 9 = 7056 |
| all but 2 | (8 × (7 / 1 – 5))3 = 4096 (8 × (7 – 5))31 = 4096 75 – 9 / 3 = 16804 (4 + 9 / 3)5 = 16807 6 × (5 – 1)7 = 98304 |
| all but 3 | (9 – 4 / 2)5 = 16807 495/2 = 16807 75 + 4 / 2 = 16809 |
| all but 4 | (60 / 5)9/3 = 1728 6 × 79/3 / 1 = 2058 6 × 7(9/3)1 = 2058 6 × 791/3 = 2058 (6 × 79/3)1 = 2058 (9 – 0)6 / 35 = 2187 96 / (3 – 0)5 = 2187 56 × 73 = 19208 |
| all but 5 | 78–36/9 = 2401 (6 × (8 / 4)9)1 = 3072 6 × (8 / 4)9/1 = 3072 6 / (1 – 4 / 8)9 = 3072 (7 – 3)8–2 / 1 = 4096 (7 / 1 – 3)8–2 = 4096 (7 – 3 / 1)8–2 = 4096 (7 – 3)8/1–2 = 4096 (7 – 3)(8–2)/1 = 4096 (8 / (7 – 3) )12 = 4096 (8 / 2)13–7 = 4096 (1 × 27 / 8)3 = 4096 ( (2 / 1)7 / 8)3 = 4096 (27/1 / 8)3 = 4096 (27 / 8)3/1 = 4096 (271 / 8)3 = 4096 (7 / 1 – 3)6 + 2 = 4098 ( (7 – 3) / 1)6 + 2 = 4098 (7 – 3)6/1 + 2 = 4098 862/9 + 7 = 4103 (63 + 1)2 = 47089 |
| all but 6 | (7 / (9 – 8) – 3)5 = 1024 (7 – 39–8)5 = 1024 (7 – 3)59–8 = 1024 5710–8 = 3249 7 × 2(9–0)/1 = 3584 7 × 29/1–0 = 3584 5 × 47 + 2 = 81930 |
| all but 7 | (7 – 3)5/(9–8) = 1024 (8 – 36 / 9)5 = 1024 463 / 92 = 1058 (215 – 8) / 9 = 3640 28/(5/3–1) = 4096 852/13 = 4096 (832–5) / 1 = 4096 832–5/1 = 4096 8(32–5)/1 = 4096 832/1–5 = 4096 8(3/1)2–5 = 4096 832/1–5 = 4096 832–51 = 4096 (8 / 1)5 / 23 = 4096 85/1 / 23 = 4096 851 / 23 = 4096 85 / 23/1 = 4096 (82 / (5 – 1) )3 = 4096 (94 – 28)1 = 6305 3 × 5621 = 9408 (3 × 562)1 = 9408 |
| all but 8 | (63 / 9)5 / 7 = 2401 (7 – 5)13 / 2 = 4096 215/3+7 = 4096 |
| all but 9 | 7(6/3)5/8 = 2401 (20 / 5)7 = 16384 270/5 = 16384 |
| 123 × 4 + 58 = 6970 | 75 – (9 / 3)8 = 10246 | 8 × 64 + 27 = 10395 | 72 × 69/3 = 10584 |
| (6 × (9 – 2) )3 / 7 = 10584 | (27 – 5)9/3 = 10648 | (48 + 7 – 5) / 6 = 10923 | 48 / 6 + 7 / 3 = 10925 |
| 5 × (6 / 2)7 + 8 = 10943 | 372 × 8 + 4 = 10956 | (7 + 90)5 / 2 = 16384 | 75 + (3 – 9) / 2 = 16804 |
| (94–3) – 2)5 = 16807 | (9 / (4 – 3) – 2)5 = 16807 | (9 – 2)5/(4–3) = 16807 | (9 – 2)5×(4–3) = 16807 |
| (9 – 2)54–3 = 16807 | (42/ (9 – 3) )5 = 16807 | (34 / 9 – 2)5 = 16807 | 75 + 2 / (4 – 3) = 16809 |
| 75×(4–3) + 2 = 16809 | 754–3 + 2 = 16809 | 75 + 92 / 4 = 16830 | 75 + 48 × 2 = 16903 |
| 8 × 94 / 3 + 6 = 17502 | 35 × 46 / 8 = 17920 | 7 × (56 / 4)3 = 19208 | 274–50 = 19683 |
| 39 + 768 = 20451 | 81 × 49–5 = 20736 | (57 + 19) / 3 = 26048 | 109 × 35 = 26487 |
| (16 – 3)4 + 9 = 28570 | (4 × (1 + 90) )5 = 32768 | (1 × (9 – 40) )5 = 32768 | (9 / 1 – 40)5 = 32768 |
| (9 – 40)5/1 = 32768 | (9 – 40)51 = 32768 | (9 – 140)5 = 32768 | (91 – 40)5 = 32768 |
| 85 + 140 = 32769 | 4 × (57 – 1) / 8 = 39062 | 2097–5 = 43681 | (651 / 3)2 = 47089 |
| 3 × (85 / 2 + 6) = 49170 | (76 × 3 + 0)2 = 51984 | (76 × 3 – 0)2 = 51984 | (76 × (3 – 0 ) )2 = 51984 |
| ( (76 – 0) × 3)2 = 51984 | (76 × 3)2–0 = 51984 | (49–5 + 3)2 = 67081 | 59–2 – 43 = 78061 |
| 5 × 647/3 = 81920 | 5 × 49–1 = 327680 | (17 + 4)8 = 390625 | 58 + 17 = 390642 |
| 843 + 6 = 592710 |
If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 6/1/12.