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?


ANSWERS

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 = 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

Two Digits
012345678
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 = 4000011 + 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 × 4344 / 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 / 552 = 25(53 – 5) / 3 = 35 + 5(5 – 4) × (44 + 44 + 4)
= 555 + 5 – 44
(Berend van der Zwaag)
6 6 × (60 / 6)6 = 60000006 + 6 = 11 + 126 = 62 + 26 × 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 = 7000000011 + 1 + 1 + 1 = 7 + 7(27 + 7) / (7 – 2) = 27 33 + 37 = (7 – 3)3 57 – 55 / 5 + 55 × 5 = 7777566 / 6 = 7776
8 8 × (80 / 8)8 = 80000000011 + 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 × 79 × 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+12345678
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+2345678
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+345678
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)

Three Digits With Lowest Digit 3
3+45678
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+5678
6 6 × 65 = 46656
7 6 × 64 = 7776
8 66 = 46648 + 8
9

Three Digits With Lowest Digit 5
5+678
7 65 = 7776
8
9

Three Digits With Lowest Digit 6
6+78
8
9

Three Digits With Lowest Digit 7
7+8
9

Eight Digits
all but012345678
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 = 65706(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 = 40968(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 = 4096893/6–0 = 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 = 2567222 = 33232930569601
(Andrew Bayly)
1210 = 61917364224
(Andrew Bayly)
(7 – 9 / 3)5 = 1024362 = 150094635296999121
(Andrew Bayly)
9 867–43 = 512 2410–8 = 576
8421 = 7056
(84 × 1)2 = 7056
3 × 210 / 4 = 76878/(5–3) = 240165 / 3 / 24 = 108
803 / 46 = 125

Nine Digits
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

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.