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, 2, 3, (and in that order) along with the operations
listed above to write expressions for all the numbers from 1 to 23.
Extra credit for those who can go past 23 (consecutively).
Click here for a worksheet
Click here for solutions to previous years
Some Solutions to the Problem:
1 = 2 + 0 + 2 - 32 = 2 + 0 * 23
3 = 2 + 0 - 2 + 3
4 = 2 + 0! - 2 + 3
5 = 2 * 0 + 2 + 3
6 = 2 - 0! + 2 + 3
7 = 2 + 0 + 2 + 3
8 = 2 + 0! + 2 + 3
9 = ((2 * 0)! + 2) * 3
10 = (2 + 0) * (2 + 3)
11 = 2 + 0! + 2^3
12 = (2 + 0) * 2 * 3
13 = [sqrt(202)] - [sqrt(3)]
14 = 2 * (0! + 2 * 3)
15 = 20 - 2 - 3
16 = 20 - [sqrt(23)]
17 = 20 - 2 - [sqrt(3)]
18 = 20 - [sqrt(2)] - [sqrt(3)]
19 = 20 + 2 - 3
20 = 20 - [sqrt(2)] + [sqrt(3)]
21 = 20 - 2 + 3
22 = 20 + [sqrt(2)] + [sqrt(3)]
23 = 20 + 2 + [sqrt(3)]
24 = 2 * 0 - 2 + [sqrt((3!)!)]
25 = 20 + 2 + 3
26 = 20 + 2 * 3
27 = 2 + 0! - 2 + [sqrt((3!)!)]
28 = 2 * 0 + 2 + [sqrt((3!)!)]
29 = 2^0 + 2 + [sqrt((3!)!)]
30 = 2 + 0 + 2 + [sqrt((3!)!)]
31 = 2 + 0! + 2 + [sqrt((3!)!)]
32 = [sqrt(20)] * (2^3)
33 = (2+0!)!^2 - 3
Here is Colin Bowey's complete list:
1 = 2+0+2-3
2 = 2+0*(2+3)
3 = (2*0)*2+3
4 = -2+0+(2*3)
5 = (2*0)+2+3
6 = (2*0)+(2*3)
7 = ((2+0)*2)+3
8 = 2+0!+2+3
9 = 2+0!+(2*3)
10 = 2+0+(2^3)
11 = 2+0!+(2^3)
12 = (2+0)*2*3
13 = (20/2)+3
14 = 20-(2*3)
15 = 20-2-3
16 = (2+0)*(2^3)
17 = -(2+0!)!+23
18 = (2+0!)*(2*3)
19 = 20+2-3
20 = -2-0!+23
21 = 20-2+3
22 = -2+0!+23
23 = (2*0)+23
24 = 2-0!+23
25 = 2+0+23
26 = 2+0!+23
27 = (2*0)+INT(SQRT(2))+INT(SQRT((3!)!))
28 = 20+(2^3)
29 = 2+(0!+2)^3
30 = 20/2*3
31 = INT(((2+0!)!)!/23)
32 = 2^(0+2+3)
33 = (2+0!)!^2-3
34 = (20*2)-3!
35 = (20*2)-INT(SQRT(SQRT((3!)!)))
36 = (2+0!)!*2*3
37 = (20*2)-3
38 = INT((20*2)-SQRT(3))
39 = (2+0!)!^2+3
40 = (2+0!+2)!/3
41 = INT(SQRT(((2+0!)!)!)+SQRT(SQRT((2^3)!)))
42 = (2+0!)!^2+3!
43 = (20*2)+3
44 = INT(SQRT(2023))
45 = INT(SQRT(SQRT((20/2)!))+SQRT(3))
46 = (2+0)*23
47 = 20+INT(SQRT(2))+INT(SQRT((3!)!))
48 = 2*(0!+23)
49 = INT(2*SQRT(SQRT((0!+2^3)!)))
50 = INT(SQRT(((2+0!)!)!)*2)-3
51 = INT((SQRT(((2+0!)!)!)*2)-SQRT(3))
52 = INT((SQRT(((2+0!)!)!)*2)-SQRT(SQRT(3)))
53 = INT(INT(SQRT(((2+0!)!)!)*2))*INT(SQRT(SQRT((SQRT(3)))))
54 = 2*((0!+2)^3)
55 = INT(SQRT(((2+0!)!)!)*2+SQRT(3))
56 = INT(SQRT(((2+0!)!)!)*2)+3
57 = INT(SQRT(SQRT(SQRT(20!)))/2/SQRT(3))
58 = 2^(0!+2)!-3!
59 = 2^(0!+2)!-INT(SQRT(SQRT((3!)!)))
60 = 20*INT(SQRT(2))*3
61 = 2^(0!+2)!-3
62 = INT(20*(SQRT(2)+SQRT(3)))
63 = ((20+INT(SQRT(2)))*3)
64 = (2+0+2)^3
65 = INT(SQRT(((2+0!)!)!*2*3))
66 = (20+2)*3
67 = (INT(202/3))
68 = INT(INT((20*SQRT(2)))*(SQRT(3!)))
69 = INT(20*2*SQRT(3))
70 = ((2+0!)!+(2^3!))
71 = INT(SQRT((INT(SQRT(SQRT(((2+0!)!)!)))!+2)))+INT(SQRT(3))
72 = (2+0+2)!*3
73 = INT(SQRT((INT(SQRT(SQRT(((2+0!)!)!)))+2)!)+3)
74 = INT(SQRT(SQRT(SQRT(SQRT(20^23)))))
75 = INT((SQRT(SQRT((20/2)!)))*SQRT(3))
76 = INT(SQRT((INT(SQRT(SQRT(((2+0!)!)!)))+2)!))+3!
77 = INT(20*(SQRT(2)+SQRT(3!)))
78 = INT(SQRT(SQRT(SQRT(20!))))-((INT(SQRT(SQRT((3!)!)))))!)
79 = INT((SQRT(SQRT(SQRT(20!))))/(2*SQRT(SQRT(SQRT(3!)))))
80 = 20*2*INT(SQRT(3!))
81 = INT(SQRT(SQRT(SQRT(20!)))/ (SQRT(2*3)))
82 = INT(202/SQRT(3!))
83 = INT(20*(SQRT(SQRT(2))+3))
84 = 20+(2^3!)
85 = INT(SQRT(202)*3!)
86 = INT((SQRT(SQRT(SQRT(20!))))*(2-SQRT(SQRT(3!))))
87 = INT((SQRT(((2+0!)!)!))*(SQRT(SQRT(SQRT(2))))*3)
88 = INT(SQRT((2+0!)!^(2+3)))
89 = INT(202/SQRT(SQRT(INT(SQRT((3!)!)))))
90 = INT(SQRT(((2+0!)!)!)+(2^3!))
91 = INT(((2+0!+2)!)/SQRT(SQRT(3)))
92 = INT(INT(INT(SQRT(SQRT(SQRT(20!)))))/2/(SQRT(SQRT(SQRT(SQRT(3))))))
93 = INT(SQRT(SQRT(SQRT(20!)))/2)-3!
94 = INT(SQRT(SQRT(SQRT(20!)))/2)-INT(SQRT(SQRT((3!)!)))
95 = INT(INT(INT(SQRT(SQRT(SQRT(20!)))))/2/SQRT(SQRT(SQRT(SQRT(SQRT(3))))))
96 = INT(SQRT(SQRT(SQRT(20!)))/2)-3
97 = INT(20*2*SQRT(3!))
98 = INT(SQRT(SQRT(SQRT(20!)))/2)-INT(SQRT(3))
99 = INT(SQRT(SQRT(SQRT((20!)))/2)*INT(SQRT(3))
100 = 20*(2+3)
Here is Milos Vukovic's complete list:
1 2*0-2+3
2 2+0*2*3
3 2*0*2+3
4 (2+0)/2+3
5 2*0+2+3
6 2*0+2*3
7 2*0!+2+3
8 2+0!+2+3
9 2+0!+2*3
10 2*0!*(2+3)
11 (2+0!)!+2+3
12 2*(0!+2+3)
13 2-0!+2*3!
14 2*(0!+2*3)
15 (2+0!+2)*3
16 2*0!*(2+3!)
17 -[sqrt(2)]+(0!+2)*3!
18 (2+0!)*2*3
19 [sqrt(2)]+(0!+2)*3!
20 20*(-2+3)
21 20-2+3
22 -2+0!+23
23 2*0+23
24 2-0!+23
25 2+0+23
26 2+0!+23
27 (2+0+2)!+3
28 20+2^3
29 2+(0!+2)^3
30 20/2*3
31 [(20-2)*sqrt(3)]
32 2^(0+2+3)
33 2^([sqrt(sqrt(((0!+2)!)!))])+[sqrt(3)]
34 [20*[sqrt(2)]*sqrt(3)]
35 ((2+0!)!)^2-[sqrt(3)]
36 ((2+0!)!)^[sqrt(2+3)]
37 20*2-3
38 20*2-[sqrt(3!)]
39 20*2-[sqrt(3)]
40 20*[sqrt(2+3)]
41 20*2+[sqrt(3)]
42 20*2+[sqrt(3!)]
43 20+23
44 [sqrt(2023)]
45 [sqrt(sqrt(sqrt(20!)))/(sqrt(2)+3)]
46 20*2+3!
47 [20*(sqrt(sqrt(sqrt(sqrt(2))))+sqrt(sqrt(3)))]
48 (2^(0!+2))*3!
49 [20*(sqrt(2)+sqrt(sqrt(sqrt(sqrt(3)))))]
50 [sqrt(((2+0!)!)!)]+([sqrt(2)]+3)!
51 [20*(sqrt(2)+sqrt(sqrt(sqrt(3))))]
52 [20*2*sqrt(sqrt(3))]
53 [sqrt(((2+0!)!)!)*[sqrt(2+3)]]
54 [20*([sqrt(2)]+sqrt(3))]
55 [(20-sqrt(2))*3]
56 [(20-sqrt(sqrt(2)))*3]
57 (20-[sqrt(2)])*3
58 [20*(sqrt(sqrt(2))+sqrt(3))]
59 2^((0!+2)!)-[sqrt(sqrt((3!)!))]
60 20*[sqrt(2)]*3
61 2^((0!+2)!)-3
62 [20*(sqrt(2)+sqrt(3))]
63 2^((0!+2)!)-[sqrt(3)]
64 2^((0!+2)!)*[sqrt(3)]
65 2^((0!+2)!)+[sqrt(3)]
66 (20+2)*3
67 2^((0!+2)!)+3
68 [2^(-0!+sqrt(sqrt(sqrt(2)))+3!)]
69 (2+0!)*23
70 2^((0!+2)!)+3!
71 [[sqrt(((2+0!)!)!)]*([sqrt(2)]+sqrt(3))]
72 (2+0+2)!*3
73 [sqrt(((2+0!)!)!)*(sqrt(2)+sqrt(sqrt(3)))]
74 [20*(2+sqrt(3))]
75 [sqrt(((2+0!)!)!)-[sqrt(2)]]*3
76 [(sqrt(((2+0!)!)!)-sqrt(2))*3]
77 [sqrt(sqrt(sqrt(20!)))/(sqrt(2)+sqrt(sqrt(sqrt(3))))]
78 [sqrt(((2+0!)!)!)*[sqrt(2)]]*3
79 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(2))+sqrt(sqrt(3)))]
80 20*[sqrt(2)+3]
81 (2+0!)^[sqrt(2)+3]
82 [202/(sqrt(3!))]
83 [[sqrt(sqrt(sqrt(20!)))]/(sqrt(sqrt(sqrt(sqrt(2))))+sqrt(sqrt(3)))]
84 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(sqrt(sqrt(2))))+sqrt(sqrt(3)))]
85 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(2))+sqrt(sqrt(sqrt(3))))]
86 [sqrt(sqrt(sqrt(20!)))/2/sqrt(sqrt(sqrt(3)))]
87 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(2))+sqrt(sqrt(sqrt(sqrt(3)))))]
88 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(sqrt(2)))+sqrt(sqrt(sqrt(3))))]
89 [(2+0!)^(sqrt(sqrt(sqrt(2)))+3)]
90 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(sqrt(sqrt(2))))+sqrt(sqrt(sqrt(3))))]
91 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(sqrt(2)))+sqrt(sqrt(sqrt(sqrt(3)))))]
92 [sqrt(sqrt(sqrt(20!)))/2/sqrt(sqrt(sqrt(sqrt(3))))]
93 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(sqrt(sqrt(2))))+sqrt(sqrt(sqrt(sqrt(3)))))]
94 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(sqrt(sqrt(sqrt(2)))))+sqrt(sqrt(sqrt(sqrt(3)))))]
95 [20*sqrt(23)]
96 [sqrt(sqrt(sqrt(20!)))]/2-3
97 [sqrt(sqrt(sqrt(20!)))/2-sqrt(3)]
98 [sqrt(sqrt(sqrt(20!)))]/2-[sqrt(3)]
99 [(2+0!)^(sqrt(sqrt(2))+3)]
100 20*(2+3)
101 202/[sqrt(3!)]
102 [(sqrt(2)+0!+sqrt(2))*sqrt((3!)!)]
103 [20*[sqrt(2)]*sqrt(sqrt((3!)!))]
104 [sqrt(((2+0!)!)!)]*(0!+3)
105 (20+[sqrt(2)])*[sqrt(sqrt((3!)!))]
106 [2^([sqrt(sqrt(((0!+2)!)!))]+sqrt(3))]
107 [sqrt(((2+0!)!)!)*(0!+3)]
108 [(20+[sqrt(2)])*sqrt(sqrt((3!)!))]
109 [(20+sqrt(sqrt(2)))*sqrt(sqrt((3!)!))]
110 [(20+sqrt(2))*sqrt(sqrt((3!)!))]
111 -(2+0!)^2+[sqrt(sqrt((3!)!))]!
112 [sqrt([sqrt(20)]!)*23]
113 [(20+2)*sqrt(sqrt((3!)!))]
114 -(2+0!)!+(2+3)!
115 [20^(-sqrt(2)+3]
116 [202/sqrt(3)]
117 [sqrt([sqrt(((2+0!)!)!)])*23]
118 -[sqrt((2+0!)!)]+(2+3)!
119 -2+0!+(2+3)!
120 20*2*3
121 2-0!+(2+3)!
122 2+0+(2+3)!
123 2+0!+(2+3)!
124 [sqrt(20)+(2+3)!]
125 (2+0!+2)^3
126 (2+0!)!+(2+3)!
127 [2*sqrt((0!+2)!)*[sqrt((3!)!)]]
128 2^(-0!+2^3)
129 [202/sqrt(sqrt(3!))]
130 [sqrt((2+0!+2)!)]+[sqrt(sqrt((3!)!))]!
131 [2*sqrt((0!+2)!)*sqrt((3!)!)]
132 (20+2)*(3!)
133 [(sqrt([sqrt(sqrt((((2+0!)!)!)))]!)+sqrt(sqrt(2)))*sqrt([sqrt(sqrt((3!)!))]!)] explanation (sq(120)+sq(sq2))*sq(120)
134 sqrt(((2+0!)!)!)/2*[sqrt([sqrt(sqrt((3!)!))]!)] explanation 26.8/2*10
135 [(sqrt([sqrt(sqrt((((2+0!)!)!)))]!)+sqrt(2))*sqrt([sqrt(sqrt((3!)!))]!)]
136 [([sqrt(sqrt(((2+0!)!)!))]!-sqrt(2))*sqrt(sqrt(sqrt(3)))]
137 [((2+0!+2)!)*sqrt(sqrt(sqrt(3)))]
138 [[sqrt(2)]+([sqrt(sqrt(((0!+2)!)!))]!)*sqrt(sqrt(sqrt(3)))]
139 [([sqrt(sqrt((((2+0!)!)!)))]!)*sqrt(sqrt(2))-3]
140 [([sqrt(sqrt((((2+0!)!)!)))]!)*sqrt(sqrt(2))-sqrt(3)]
141 [(sqrt([sqrt(sqrt((((2+0!)!)!)))]!)+2)*sqrt([sqrt(sqrt((3!)!))]!)] explanation (sq(120)+2)*sq(120)
142 (sqrt(sqrt(2))+0)*((2+3)!)
143 [([sqrt(sqrt((((2+0!)!)!)))]!)*sqrt(sqrt(2))+[sqrt(3)]]
144 [sqrt(20)]!*2*3
145 [2^(0+sqrt(sqrt(2))+3!)]
146 [[sqrt(sqrt(((2+0!)!)!))]!+sqrt((2*3)!)]
147 [sqrt(sqrt(((2+0!)!)!))]!+[sqrt(2)]+[sqrt((3!)!)]
148 [20*(sqrt(2)+3!)]
149 [(20-2)^sqrt(3)]
150 [(sqrt(sqrt(sqrt(20!)))-[sqrt(2)])/sqrt(sqrt(3))]
151 [(sqrt(sqrt(sqrt(20!)))+[sqrt(2)])/sqrt(sqrt(3))]
152 [(sqrt(sqrt(sqrt(20!)))+2)/sqrt(sqrt(3))]
153 [202/sqrt(sqrt(3))]
154 [sqrt((((2+0!)!)!)/sqrt(sqrt(sqrt(2))))*(3!)]
155 [[sqrt(((2+0!)!)!)]/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2)))))))*3!]
156 [sqrt(((2+0!)!)!)]*2*3
157 ((2+0!+2)!)*sqrt(sqrt(3))
158 [sqrt(sqrt(sqrt((20!)/(2*3))))]
159 [2^((0!+2)!+sqrt(sqrt(3)))]
160 20*(2^3)
161 [202/(sqrt(sqrt(sqrt(3!))))]
162 [sqrt(sqrt(sqrt(20!)))*sqrt(2)/sqrt(3)]
163 [(20^2)/sqrt(3!)]
164 [(20-[sqrt(2)])^sqrt(3)]
165 [sqrt(sqrt(((2+0!)!)!))*2^[sqrt(sqrt((3!)!))]]
166 [sqrt(sqrt(sqrt(20!)))-sqrt(sqrt(2))*sqrt((3!)!)]
167 [sqrt(sqrt(sqrt(20!)/[sqrt(2+3)]))]
168 [sqrt(2)*(-0!+(2+3)!)]
169 sqrt(2)*(0+2+3)!
170 [2^(0+sqrt(2)+3!)]
171 [sqrt(sqrt(sqrt(20!)))-sqrt((2*3)!)]
172 [sqrt(sqrt(((2+0!)!)!))]!+2*[sqrt((3!)!)]
173 [[sqrt(sqrt(((2+0!)!)!))]!+2*sqrt((3!)!)]
174 [sqrt(sqrt(sqrt((20+2)!)))/sqrt(3!)]
175 [202-sqrt((3!)!)]
176 [202-[sqrt((3!)!)]]
177 [a*sqrt(b+c)] where a = 120 = [sqrt(sqrt(((2+0!)!)!))]!, b = sqrt(sqrt(sqrt(sqrt(2)))), c = sqrt(sqrt(sqrt(3)))
178 [a*sqrt(b+c)] where a = 120 = [sqrt(sqrt(((2+0!)!)!))]!, b = sqrt(sqrt(2)), c = sqrt(sqrt(sqrt(sqrt(sqrt(3)))))
179 [20^sqrt([2+sqrt(3)])]
180 [202/sqrt(sqrt(sqrt(sqrt(3!))))]
181 [[sqrt(sqrt(((2+0!)!)!))]*(2^sqrt(sqrt((3!)!)))] explanation 5*2^sq(sq(720))
182 [sqrt(sqrt(sqrt((20!)/[sqrt(2+3)])))] 183 [a*sqrt(b+c)] where a = 120 = [sqrt(sqrt(((2+0!)!)!))]!, b = sqrt(sqrt(sqrt(sqrt(sqrt(2))))), c = sqrt(sqrt(3))
184 [a*sqrt(b+c)] where a = 120 = [sqrt(sqrt(((2+0!)!)!))]!, b = sqrt(sqrt(sqrt(sqrt(2)))), c = sqrt(sqrt(3))
185 -2+[([sqrt(sqrt(((0!+2)!)!))]!)*sqrt(sqrt(3!))]
186 [(((2+0!)!)!)/(sqrt(2)+sqrt(3!))]
187 [sqrt(sqrt(sqrt(20!)))-sqrt((2+3)!)]
188 [202/sqrt(sqrt(sqrt(sqrt(3))))]
189 [sqrt(sqrt(20^([sqrt(2)]+3!)))]
190 [sqrt(sqrt(sqrt(20!)))-2^3]
191 [sqrt(sqrt(sqrt(20!)))]-[sqrt(2)]-3!
192 [sqrt(sqrt(sqrt(20!)))]-2*3
193 [sqrt(sqrt(sqrt(20!)))]+[sqrt(2)]-3!
194 [sqrt(sqrt(sqrt(20!)))]-[sqrt(2)]-3
195 [sqrt(sqrt(sqrt(20!)))]-[sqrt(2)]*3
196 202-3!
197 202-[sqrt(sqrt((3!)!))]
198 [sqrt(sqrt(sqrt((-2-0!+23)!)))]
199 202-3
200 202-[sqrt(3!)]
201 202-[sqrt(3)]
202 202*[sqrt(3)]
203 202+[sqrt(3)]
204 202+[sqrt(3!)]
205 202+3
206 [sqrt(sqrt(sqrt(20!)))+2^3]
207 202+[sqrt(sqrt((3!)!))]
208 202+3!
209 [202*sqrt(sqrt(sqrt(sqrt(sqrt(3)))))]
210 202*sqrt(sqrt(sqrt(sqrt([sqrt(3!)]))))
211 [(20+2)^sqrt(3)]
212 [2^((0!+2)!+sqrt(3))]
213 [202*sqrt(sqrt(sqrt(sqrt(sqrt(3!)))))]
214 [sqrt(((2+0!)!)!)*2^3]
215 [[sqrt(sqrt(sqrt(20!)))]*sqrt(sqrt(sqrt([sqrt(2*3)])))]
216 ((2+0+[sqrt(2)])!)^3
Correctly solved by:
| 1. Colin Bowey ** (100) | Beechworth, Victoria, Australia |
| 2. K. Sengupta ** (46) | Calcutta, India |
| 3. Ivy Joseph ** (30) | Pune, Maharashtra, India |
| 4. Davit Banana | Istanbul, Turkey |
| 5. Dr. Hari Kishan ** (28) |
D.N. College, Meerut, Uttar Pradesh, India |
| 6. Milos Vukovic ** (216) | Budapest, Hungary |
| 7. Kelly Stubblefield ** (30) | Mobile, Alabama |
| 8. Ritwik Chaudhuri ** (30) | Santiniketan, West Bengal, India |
| 9. Tabor ** (24) |
Muckleshoot Tribal School, Auburn, Washington |
** Extra Credit