Before I retired, I would give my students the following challenge:
Write expressions for all the numbers from 1 to 100 using only the digits in the current year in order
and using the operations +, -, x, ÷ (or / for divided by),
^ (raised to a power), sqrt (square root),
! (factorial),
and int (or [] for greatest integer function), along with grouping symbols.
So, the first problem of the new year is to use only the digits 2, 0, 1, 9, (and in that order) along with the operations
listed above to write expressions for all the numbers from 0 to 21.
Extra credit for those who can go past 21 (consecutively).
Click here for a worksheet
Click here for solutions to previous years
James Alarie has written a program that solves the New Year Challenge for all the years from 2019 through 2199 (yes, that is the next century). I thought that
I should mention that because I am not planning to be alive then.
Here are my solutions to the Problem:
0 = 2 * 0 * 1 * 91 = 2^0 * 1^9 or 20 - 19
2 = 2 + 0 * 19
3 = 2 + 0 + 1^9
4 = 2 + 0! + 1^9
5 = 20 / (1 + sqrt(9))
6 = 2 + 0 + 1 + sqrt(9)
7 = 2 + 0! + 1 + sqrt(9)
8 = [sqrt(20)] + 1 + sqrt(9)
9 = [sqrt(20)] - 1 + (sqrt(9))!
10 = 2 * 0 + 1 + 9
11 = [sqrt(20)] + 1 + (sqrt(9))!
12 = 2 + 0 + 1 + 9
13 = [sqrt(20)] + 1 * 9
14 = [sqrt(20)] + 1 + 9
15 = 20 + 1 - (sqrt(9))!
16 = -2 - 0! + 19
17 = -2 + 0 + 19
18 = -(2^0) + 19 or -2 + 0! + 19
19 = 2 * 0 + 19 or 20 - 1^9
20 = 20 * 1^9
21 = 20 + 1^9 or 2 + 0 + 19
22 = [201 / 9]
23 = 20 + 1 * sqrt(9)
24 = 20 + 1 + sqrt(9)
25 = [sqrt(20)]! + 1^9
26 = [sqrt(20)]! - 1 + sqrt(9)
27 = (2 + 0 + 1) * 9
28 = [sqrt(20)]! + 1 + sqrt(9)
29 = [sqrt(20)]! - 1 + (sqrt(9))! or 20 * 1 + 9
30 = [sqrt(20)]! * 1 + (sqrt(9))! or 20 + 1 + 9
31 = [sqrt(20)]! + 1 + (sqrt(9))!
32 = [sqrt(20)]! - 1 + 9 or [sqrt(20)] * (-1 + 9)
33 = [sqrt(sqrt(sqrt(20!))] * 1 / (sqrt(9)!) or [201 / ((sqrt(9))!)]
34 = [sqrt(20)]! + 1 + 9
35 = [sqrt(20)]! + [sqrt(sqrt(sqrt(sqrt(19!))))]
36 = (2 + 0! + 1) * 0
37 = (2 + 0!)! + 1 + [sqrt(sqrt(sqrt(sqrt( ([sqrt(sqrt(9!))])! ))))]
38 = (2 + 0) + 19
39 = 20 + 19
40 = [sqrt(20)] * (1 + 9)
41 = -2 + [(sqrt(0! + 1)) (sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))])))))]
42 = (20 + 1) * ([sqrt((sqrt(9))!)]) or [sqrt(201 * 9)]
43 = [sqrt(20)]! + 19
44 = [sqrt(20)] * ([sqrt(sqrt(sqrt(sqrt(19!))))])
45 = ([sqrt(20)] + 1) * 9
46 = ([sqrt(20)]! - 1) * [sqrt((sqrt(9))!)]
47 = [sqrt(20)]! - 1 + [sqrt(sqrt(9!))]
48 = [sqrt(20)]! * 1 + [sqrt(sqrt(9!))]
49 = [sqrt(20)]! + 1 + [sqrt(sqrt(9!))]
50 = (2 + 0) * (1 + [sqrt(sqrt(9!))])
51 = (20 + 1) + [sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]
52 = 2 * 0! * 1 * [sqrt([(sqrt(9))!]!)]
53 = -([sqrt(sqrt(sqrt(sqrt(20!))))]) + [sqrt(sqrt(sqrt(sqrt(sqrt(sqrt([sqrt(sqrt( [sqrt(sqrt(sqrt(sqrt(19!))))]!))]! ))))))]
54 = ((2 + 0!)!) * 1 * 9
55 = -([sqrt(20)]!) + [sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))]
56 = 2 * (0! + 1 + [sqrt([(sqrt(9))!]!)])
57 = (2 + 0!) * 19 or (20 - 1) * (sqrt(9))
58 = (2 + 0) * (-1 + ([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))
59 = -20 + [sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))]
60 = ((2 + 0!)!) * (1 + 9) or 20 * 1 * sqrt(9) or [(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt(20!))))]!))) * 1 / 9]
61 = [sqrt(sqrt(sqrt(sqrt([sqrt(20)]!))))] + 1 + [sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]
62 = [[sqrt(sqrt(sqrt(20!)))] / (sqrt(1 + 9))]
63 = ((2 + 0!)! + 1) * 9 or (20 + 1) * (sqrt(9))
64 = -2 - 0! + [sqrt(sqrt(sqrt(sqrt(sqrt(sqrt([sqrt(sqrt( [sqrt(sqrt(sqrt(sqrt(19!))))]!))]! ))))))]
65 = -([sqrt(sqrt(sqrt(sqrt(20!))))]) + [sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))]
66 = [sqrt(sqrt(sqrt(20!)))] * 1 / sqrt(9)
67 = 201 / sqrt(9)
68 = 2 - 1 + [sqrt(sqrt(sqrt(sqrt(sqrt(sqrt([sqrt(sqrt( [sqrt(sqrt(sqrt(sqrt(19!))))]!))]! ))))))]
69 = [sqrt(sqrt(sqrt([sqrt(201)]!)))] * (sqrt(9))
70 = [([sqrt(sqrt(sqrt(20!)))]) / (sqrt(-1 + 9)]
71 = [sqrt(20)] + [sqrt(sqrt(sqrt(sqrt(sqrt(sqrt([sqrt(sqrt( [sqrt(sqrt(sqrt(sqrt(19!))))]!))]! ))))))]
72 = (2 + 0 + 1) * ([sqrt(sqrt(9!))])
73 = [sqrt(sqrt(sqrt(sqrt([sqrt(20)]!))))] * 1 + [sqrt(sqrt([sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))])]!))]
74 = [sqrt(sqrt(sqrt(sqrt([sqrt(20)]!))))] + 1 + [sqrt(sqrt([sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))])]!))]
75 = -([sqrt(20)]) + ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))])
76 = [sqrt(20)] * 19
77 = -2 + 0 + ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))])
78 = -2 + 0! + ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))])
79 = (2 - 0!) * ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))])
80 = 2 - 0! + ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))]) or (20) * (1 + sqrt(9))
81 = 2 + 0 + ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))])
82 = 2 + 0! + ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))])
83 = [sqrt(20)] + ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))])
84 = [sqrt(sqrt(sqrt(sqrt(20!))))] * 1 * ((sqrt(9))!)
85 = [((sqrt(201)) * ((sqrt(9))!))]
86 = ((2 + 0) * 1) * [sqrt(sqrt([sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))])]!))]
87 = -20 + 1 + [sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))]
88 = (2 + 0) * (1 + [sqrt(sqrt([sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))])]!))])
89 = int(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt((201)!)))))))) * sqrt(9))
90 = (2 + 0 + 1) * ([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))])
91 = [sqrt(sqrt(sqrt(20!)))] - 1 - ([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))])
92 = [sqrt(sqrt(sqrt(20!)))] * 1 - ([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))])
93 = [sqrt(sqrt(sqrt(20!)))] + 1 - ([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))]) or [sqrt(sqrt(sqrt(sqrt(20!))))] + [sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))]
94 = int(sqrt(sqrt(2)) * sqrt(sqrt(((0)! + 1 + 9)!)))
95 = 201 - [sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(sqrt(sqrt([sqrt(sqrt(9!))]))))]))))]
96 = (2 + 0! + 1) * [sqrt(sqrt(9!))]
97 =
98 = int((sqrt(sqrt(sqrt(sqrt(20!))))) * (1 + (sqrt(9)!) or int(int(sqrt(20 + 1)) * sqrt(sqrt((9)!)))
99 = 20 + ([sqrt(sqrt(([sqrt(sqrt(sqrt(sqrt(19!))))]!)))])
100 = [(201) / [sqrt((sqrt(9))!)]] or 20 * (-1 + (sqrt(9))!)
Brooks Garris filled in the missing 97:
97 = int( sqrt( int(sqrt(int(sqrt(((2+0!)!)!)))) x int(sqrt((1+9)!)) ) )
Click here for a printable version of my answers
Correctly solved by:
| 1. James Alarie *** (51 consecutive) | Flint, Michigan |
| 2. Rob Miles (21 consecutive) | Northbrook, Illinois |
| 3. Brooks Garris *** (46 consecutive) |
South Robeson High School Rowland, North Carolina |
| 4. Ivy Joseph *** (30 consecutive) | Pune, Maharashtra, India |
| 5. Kaley St. Peter (21 consecutive) |
Delta High School, Delta, Colorado |
| 6. Brijesh Dave (21 consecutive) | Mumbai City, Maharashtra, India |
| 7. Caleb Frazier (21 consecutive) |
Delta High School, Delta, Colorado |
*** solved the extra credit (more than 21 consecutive answers).