# Problem of the Month (June 2012)

If 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?

Solutions were received from Bryce Herdt, Andrew Bayly, Jon Palin, and Berend van der Zwaag.

Here are the shortest-known solutions:

One Digit
 0 0 + 0 = 0 1 1 / 1 = 111 = 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

Two Digits
 0 1 2 3 4 5 6 7 8 1 1001+1 = 10000 2 (2 × 2)2 + 22 = 2022×2 + 22 = 20 112 = 121 3 33 = 30 – 3 / 30(3 / 30)3 = 30 – 333+0/3 = 30 – 333–0/3 = 30 – 333–0–0 = 30 – 333–03 = 30 – 333/30 = 30 – 333×30 = 30 – 33330 = 30 – 3(33 + 3) / 30 = 30(33 + 3)30 = 30 113 = 133133 – 1 = (3 – 1) × 13 2 × 222 = 322 × 22+2 = 322 × 22×2 = 322 × (2 + 2)2 = 322 × (2 × 2)2 = 32(2 × 2)3 / 2 = 3223+3 / 2 = 322 × 2 × 23 = 3233 – 22 = 23 4 4 × (40 / 4)4 = 40000 11 + 1 = 4 + 4 + 4 / 111 + 1 = 4 + (4 + 4) / 111 + 1 = (4 + 4 + 4) / 111 + 1 × 1 = 4 + 4 + 411 = 4 + 4 + (4 – 1) / 1 22 + 24 / 2 = 4 × 4 344 / 43 = 43 / (4 + 4) 5 5 / 50 + 5 = 50 / 550 × 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 + 366/3 = 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 – 99 × 9 = 90 × (90 – 9)9 × 990 = 90 – 99 × 9 = 90 – 9 / 90(9 × 9)90 = 90 – 99 + 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

Three Digits With Lowest Digit 0
 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

Three Digits With Lowest Digit 1
 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)

Three Digits With Lowest Digit 2
 2+ 3 4 5 6 7 8 4 224 – 322 = 233232(Andrew Bayly) 5 6 52×2 = 625522 = 625 7 8 (82 – 2)3 = 238328(Andrew Bayly) 9 (94 – 4)2 = 42994249(Andrew Bayly)

Three Digits With Lowest Digit 3
 3+ 4 5 6 7 8 5 6 (33 – 4)3×3–4 = 6436343(Andrew Bayly) 7 73 = 343 67 = 36 × 7776 8 9

Three Digits With Lowest Digit 4
 4+ 5 6 7 8 6 6 × 65 = 46656 7 6 × 64 = 7776 8 66 = 46648 + 8 9

Three Digits With Lowest Digit 5
 5+ 6 7 8 7 65 = 7776 8 9

Three Digits With Lowest Digit 6
 6+ 7 8 8 9

Three Digits With Lowest Digit 7
 7+ 8 9

Eight Digits
 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 = 2562418/9 = 576 4 (35 / 81)6 = 729 381 + 9 = 6570 6(5–0)/1 / 8 = 97265/1–0 / 8 = 972651–0 / 8 = 9726(5–0)1 / 8 = 972610–5 / 8 = 972 5 (78 – 43)2 = 196 (27 / 8)3 = 4096 8(7–3)/1 = 409687/1–3 = 4096 786/9 = 2401 6 8(7–4)3/9 = 512(83 / 4)9/7 = 512 3151 = 14348907(Andrew Bayly) (8 / (9 – 7) )5 = 10245 × 47 = 81920 (9 – 0)4 / 3 = 2187 7 95×8/16 = 243834/9–6 = 512(46/3 / 8)9 = 512( (6 / 3)4 / 8)9 = 512 215 / 8 = 4096 893/6–0 = 5128(9–0)3/6 = 512 95/(10–8) = 243 8 (54 / 6)31 = 729(54 / 61)3 = 729(541 / 6)3 = 729361×(5–4) = 7293(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 = 5768421 = 7056(84 × 1)2 = 7056 3 × 210 / 4 = 768 78/(5–3) = 2401 65 / 3 / 24 = 108803 / 46 = 125

Nine Digits
 all but 0 694/2 / 3 = 1587(78/4 – 5)2 = 1936(72 – 5)8/4 = 1936354/6 / 9 = 2187(9 / (6 – 5) )4 / 3 = 218794/(6–5) / 3 = 21875927–6 = 3481723×6/9 = 5184(83 – 9)21 = 547698/2 + 13 = 6574(3 × 76)2 = 51984 all but 1 853–72 = 4096823/(7–5) = 4096(84 × 3)2 / 9 = 7056 all but 2 (8 × (7 / 1 – 5))3 = 4096(8 × (7 – 5))31 = 409675 – 9 / 3 = 16804(4 + 9 / 3)5 = 168076 × (5 – 1)7 = 98304 all but 3 (9 – 4 / 2)5 = 16807495/2 = 1680775 + 4 / 2 = 16809 all but 4 (60 / 5)9/3 = 17286 × 79/3 / 1 = 20586 × 7(9/3)1 = 20586 × 791/3 = 2058(6 × 79/3)1 = 2058(9 – 0)6 / 35 = 218796 / (3 – 0)5 = 218756 × 73 = 19208 all but 5 78–36/9 = 2401(6 × (8 / 4)9)1 = 30726 × (8 / 4)9/1 = 30726 / (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 = 4098862/9 + 7 = 4103(63 + 1)2 = 47089 all but 6 (7 / (9 – 8) – 3)5 = 1024(7 – 39–8)5 = 1024(7 – 3)59–8 = 10245710–8 = 32497 × 2(9–0)/1 = 35847 × 29/1–0 = 35845 × 47 + 2 = 81930 all but 7 (7 – 3)5/(9–8) = 1024(8 – 36 / 9)5 = 1024463 / 92 = 1058(215 – 8) / 9 = 364028/(5/3–1) = 4096852/13 = 4096(832–5) / 1 = 4096832–5/1 = 40968(32–5)/1 = 4096832/1–5 = 40968(3/1)2–5 = 4096832/1–5 = 4096832–51 = 4096(8 / 1)5 / 23 = 409685/1 / 23 = 4096851 / 23 = 409685 / 23/1 = 4096(82 / (5 – 1) )3 = 4096(94 – 28)1 = 63053 × 5621 = 9408(3 × 562)1 = 9408 all but 8 (63 / 9)5 / 7 = 2401(7 – 5)13 / 2 = 4096215/3+7 = 4096 all but 9 7(6/3)5/8 = 2401(20 / 5)7 = 16384270/5 = 16384

Ten Digits
 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.