Problem of the Month (April 2004)

Strobogrammatic numbers (SNs) are numbers that are the same when viewed upside down: 0, 1, 8, 11, 69, 88, 96, .... These are sequence 000787 at the Encyclopedia of Integer Sequences. This month we consider Strobogrammatic Expressions (SEs): mathematical expressions that describe the same number when viewed upside down. Only the digits 0, 1, 6, 8, and 9, addition, subtraction, multiplication, division, exponentiation, and parentheses are allowed. Here are some simple examples: 91-8/8-16, 68+68+61, and 9(9-6), which describe the numbers 74, 197, and 729 respectively.

Every number can be represented by a SE of the form 1+1+1+.... What are the shortest (in terms of the minimum number of symbols used) SEs that represent the integers from 1 to 100? We are particularly interested in shortest SEs that do not use any SNs.

Joseph DeVincentis and Richard Sabey sent SEs for 1-100, many of which beat my best.

Jeremy Galvagni found lots of SEs involving exponents.

Philippe Fondanaiche sent alternatives to some shortest expressions.

Gavin Theobald improved some expressions.

Shortest Known Strobogrammatic Expressions
NumberSymbolsShortest SE
111
231+1
339-6
459-6+1
5561-19
6269 (GT)
7469+1 (GT)
818
938+1
916
10591+16
8+1+1
916+1
11211
12411+1
1111 (GT)
13611+1+1
14469+8 (GT)
1536+9
1638+8
1758+8+1
8+916
69+11 (GT)
18518181
19411+8
20611+8+1
21589-68
22511+11
2356+8+9
2458+8+8
2578+8+8+1
8+916+8
69+8+11 (GT)
99-8-66 (GT)
2666+11+9
2768+11+8
2888+11+1+8
8+11+916
69+11+11 (GT)
29789-68+8
3076+6+9+9
11+8+11
191-161
696-666 (GT)
3176+8+8+9
3278+8+8+8
99-1-66
33599-66
34799-66+1
61-8-19 (JD)
3588+8+8+11

NumberSymbolsShortest SE
36561+19 (GT)
69*69 (GT)
37761+19+1 (GT)
69*69+1 (GT)
3896+6+8+9+9
11+8+8+11
101+18-81 (JD)
39899-66+69 (GT)
4088(61-19)
41761-1-19
42561-19
43761-19+1
44761+19+8 (GT)
69*69+8 (GT)
4579*9-6*6
4666/9*69 (JD)
4786/9*69+1 (JD)
69*69+11 (GT)
48469*8 (GT)
49669*8+1 (GT)
50761-19+8
5176*9+6-9
52881-11-18
537611-899 (GT)
5436*9
5556*9+1
56669*8+8 (GT)
5776*9+9-6
618+819 (GT)
58869+6-8-9 (GT)
59769*8+11 (GT)
60669+6*9 (GT)
61561119
6256*9+8
63581-18
6438*8
6558*8+1
66566199
69*11 (GT)
677681-189
661+199
66199+1
8*8+9-6
91-8-16 (JD)
69*11+1 (GT)
68568189
69269
70469+1
1961 (GT)
7161+69+1

NumberSymbolsShortest SE
7258*8+8 (RS)
7378*8+8+1 (RS)
74791-1-16
66199+8
111*6/9 (GT)
69*11+8 (GT)
75591-16
69+69 (GT)
76791-16+1
68189+8
69+1961 (GT)
77469+8
78669+8+1
69+916
7979+8*8+6 (GT)
819+618 (GT)
80569+11
61+19 (PF)
81581118
827811+118
81118+1
83791-16+8
69+69+8 (GT)
8466+69+9
85688+6-9
86586198
877861+198
86198+1
88288
89488+1
1881 (GT)
9061+88+1
91591116
927911+116
91116+1
86+69-0 (GT)
93696+6-9
94569+88 (GT)
957961-196 (GT)
69+1881 (GT)
96296
97496+1
1691 (GT)
98598186
99588+11
99166
81+18 (PF)
100788+11+1
99166+1
81+1+18 (PF)
1181+18 (GT)

Richard Sabey sent solutions for many other numbers less than 1000 as well.

Joseph DeVincentis also sent some SEs that are different expressions upside down. Here are some short primitive expressions that can be used to manufacture other examples.

NumberSymbolsShortest Non-Symmetric SE
13161
10881+10+16
16861+9+16
176981-81
628811+0-19
87898+0-111
100898+18+19

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/23/06.