January 2022
Problem of the Month

The New Year Challenge



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, 2, (and in that order) along with the operations listed above to write expressions for all the numbers from 1 to 22.

Extra credit for those who can go past 22 (consecutively).

Click here for a worksheet

Click here for solutions to previous years


Some Solutions to the Problem:

Here is Milos Vukovic's complete list:
1 2-0-2/2
2 2+0*22
3 2+0+2/2
4 2+0*2+2
5 20/(2+2)
6 2+0+2+2
7 2+0!+2+2
8 2*0!*2*2
9 20/2-[sqrt(2)]
10 2^(0!+2)+2
11 20/2+[sqrt(2)]
12 20/2+2
13 [sqrt(202)]-[sqrt(2)]
14 ((2+0!)!)*2+2
15 [sqrt(202)]+[sqrt(2)]
16 [sqrt(20)]^([sqrt(2)]+[sqrt(2)])
17 [sqrt(20)]^2+[sqrt(2)]
18 20-[sqrt(2)]-[sqrt(2)]
19 20-2/2
20 20-2+2
21 20+2/2
22 20+[sqrt(2)]+[sqrt(2)]
23 20+[sqrt(2)]+2
24 20+2+2
25 (2+0!+2)^2
26 (2+0!*2)!+2
27 (2+0!)^(2+[sqrt(2)])
28 [sqrt(202)]*2
29 [sqrt(sqrt(sqrt(sqrt(20!))))]*2+[sqrt(2)]
30 [sqrt(sqrt(sqrt(sqrt(20!))))]*2+2
31 [sqrt(sqrt(sqrt(sqrt(sqrt((20*([sqrt(2)]+[sqrt(2)]))!)))))]
32 2^(0!+2+2)
33 [sqrt(sqrt(sqrt(sqrt(sqrt((20*2)!)))))]+2
34 ((2+0!)!)^2-2
35 ((2+0!)!)^2-[sqrt(2)]
36 (2+0!)!*(2+[sqrt(2)])!
37 [sqrt((((2+0!)!)!)*2-2)]
38 (20-[sqrt(2)])*2
39 20*2-[sqrt(2)]
40 20*2*[sqrt(2)]
41 20*2+[sqrt(2)]
42 20*2+2
43 [sqrt(sqrt(sqrt(sqrt([sqrt(((2+0!)!)!)]!))))]-2-[sqrt(2)]
44 [sqrt(sqrt(sqrt(sqrt([sqrt(((2+0!)!)!)]!))))]-[sqrt(2)]-[sqrt(2)]
45 [sqrt(sqrt(sqrt(sqrt([sqrt(((2+0!)!)!)]!))))]-2/2
46 [sqrt(sqrt(sqrt(sqrt([sqrt(((2+0!)!)!)]!))))]-2+2
47 [sqrt(sqrt(sqrt(sqrt([sqrt(((2+0!)!)!)]!))))]+2/2
48 [sqrt(sqrt(sqrt(sqrt([sqrt(((2+0!)!)!)]!))))]+[sqrt(2)]+[sqrt(2)]
49 [sqrt(sqrt(sqrt(sqrt([sqrt(((2+0!)!)!)]!))))]+2+[sqrt(2)]
50 [sqrt(((2+0!)!)!)]*2-2
51 [sqrt(((2+0!)!)!)]*2-[sqrt(2)]
52 [sqrt(((2+0!)!)!)]*([sqrt(2)]+[sqrt(2)])
53 [sqrt(((2+0!)!)!)]*2+[sqrt(2)]
54 [sqrt(((2+0!)!)!)]*2+2
55 [(sqrt(((2+0!)!)!)+sqrt(sqrt(sqrt(2))))*2]
56 [sqrt(((2+0!)!)!)+2]*2
57 [(sqrt(((2+0!)!)!)+2)*2]
58 [sqrt(((2+0!)!)!)*2*sqrt(sqrt(sqrt(2)))]
59 [sqrt(((2+0!)!)!*sqrt((2+2)!))]
60 ((2+0!+2)!)/2
61 [sqrt(((2+0!)!)!)*(sqrt(sqrt(sqrt(2)))+sqrt(sqrt(2)))]
62 2^((0!+2)!)-2
63 2^((0!+2)!)-[sqrt(2)]
64 2^((0!+2)*2)
65 2^((0!+2)!)+[sqrt(2)]
66 2^((0!+2)!)+2
67 [sqrt(((2+0!)!)!)*(sqrt(sqrt(sqrt(2)))+sqrt(2))]
68 [20*(2+sqrt(2))]
69 [(20^sqrt(2))*[sqrt(2)]]
70 [(20^sqrt(2))+sqrt(2)]
71 [(20^sqrt(2))+2]
72 (2+0!)*((2+2)!)
73 [sqrt(sqrt((((2+0!)!)*2)!))/2]
74 [(20+[sqrt(2)])^sqrt(2)]
75 [20^(sqrt(2))*sqrt(sqrt(sqrt(2)))]
76 [(20+sqrt(2))^sqrt(2)]
77 [sqrt(((((2+0!)!)+[sqrt(2)])!)*sqrt(sqrt(2)))]
78 [sqrt(((2+0!)!)!)]*(2+[sqrt(2)])
79 [(20+2)^sqrt(2)]
80 20*(2+2)
81 (2+0!)^(2+2)
82 [sqrt(sqrt(sqrt(20!)))/([sqrt(2)]+sqrt(2))]
83 [sqrt(sqrt(sqrt(20!)))/2/sqrt(sqrt(2))]
84 [((2+0!+2)!)/sqrt(2)]
85 [2^([sqrt(sqrt(((0!+2)!)!))]+sqrt(2))]
86 [[sqrt(sqrt(sqrt(20!)))]/(sqrt(sqrt(sqrt(2)))+sqrt(sqrt(2)))]
87 [sqrt(sqrt(sqrt(20!)))/(sqrt(sqrt(sqrt(2)))+sqrt(sqrt(2)))]
88 [(20*sqrt(sqrt(2)))^sqrt(2)]
89 [sqrt(sqrt(sqrt(20!)))/sqrt(sqrt((2+2)!))]
90 [(2^((0!+2)!))*sqrt(2)]
91 [sqrt(sqrt(sqrt(20!)))/2/sqrt(sqrt(sqrt(2)))]
92 [[sqrt(sqrt(sqrt(20!)))]/(sqrt(sqrt(sqrt(2)))+sqrt(sqrt(sqrt(sqrt(2)))))]
93 [(sqrt(((2+0!)!)!)-2)^sqrt(2)]
94 [[sqrt(sqrt(sqrt(20!)))]/2/sqrt(sqrt(sqrt(sqrt(2))))]
95 [sqrt(sqrt(sqrt(20!)))/2/sqrt(sqrt(sqrt(sqrt(2))))]
96 [sqrt(20)]!*2*2
97 [sqrt(sqrt(sqrt(20!)))/2/sqrt(sqrt(sqrt(sqrt(sqrt(2)))))]
98 [sqrt(sqrt(sqrt(20!)))/2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2))))))]
99 [sqrt(sqrt(sqrt(20!)))/sqrt(2)/sqrt(2)]
100 [((2+0!+2)!)/sqrt(sqrt(2))]


Here is Ritwik Chaudhuri's complete list:
1=2×0+2÷2
2=(2+0+2)÷2
3=2+0+2÷2
4=2×0+2+2
5=(2+0!)!−(2÷2)
6=2+0+2+2
7=2+0!+2+2
8=(2+0)×(2+2)
9=(2+0!)!+2+𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(2))
10=(2+0!)!+2+2
11=(2+0!)^2+2
12=20÷2+2
13=(2+0!)!×2+𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(2))
14=(2+0!)!×2+2
15=𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(((2+0!)!)!)!)÷2+2
16=(2+0+2)^2
17=20−𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(2))−2
18=((2+0!)^2)×2
19=20−2÷2
20=−2+0+22
21=20+2÷2
22=2×0+22
23=20+2+𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(2))
24=20+2+2
25=2+0!+22
26=2+(0+2+2)!
27=(2+0!)^(2+𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(2))
28=(2+0!)!+22
29=𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(((2+0!)!)!))+2+𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(2))
30=𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(((2+0!)!)!))+2+2
31=2−0!+𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(((2+2)!)!)))))
32=2^(0!+2+2)
33=2+0!+𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(((2+2)!)!)))))
34=((2+0!)!)^2-2
35=((2+0!)!)^2-int(sqrt(2))
36=(20−2)×2
37=((2+0!)!)^2+int(sqrt(2))
38=20×2−2
39=(20×2)−𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(2))
40=𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡((20+2)!)))))×2
41=(20×2)+𝑖𝑛𝑡(𝑠𝑞𝑟𝑡(2))
42=20×2+2
43=int(2×(0!+ 𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑠𝑞𝑟𝑡(𝑓𝑎𝑐𝑡𝑜𝑟𝑖𝑎𝑙(22)))))))
44=2×(0+22)


Here is Colin Bowey's complete list:
1 =(2*0)+(2/2)
2 =-20+22 = 2+0+2-2
3 =2+0!+2-2
4 =(2*0)+2+2
5 =20/(2+2)
6 =2+0+2+2
7 =2+0!+2+2
8 =(2+0)*2*2
9 =((2+0!)!/2)^2
10 =(2+0!+2)*2
11 =((2+0!)^2)+2
12 =(20/2)+2
13 =((2+0!)!)*2)+INT(SQRT(2))
14 =((2+0!)!*2)+2
15 =INT(SQRT(((2+0!)!)!))/2)+2
16 =20-2-2
17 =20-2-INT(SQRT(2))
18 =((2+0!)^2)*2
19 = -(2+0!)+22
20 =20+2-2
21 =20+(2/2)
22 =(2*0)+22
23 =2-0!+22
24 =2+0+22 = (2+(0*2)+2)!
25 =((2+0!)+2)^2
26 =2+0+(2+2)!
27 =2+0!+(2+2)!
28 =(2+0!)!+22
29 = INT((SQRT(((2+0!)!)!))/SQRT(SQRT(SQRT(2))))*SQRT(SQRT(2)))
30 =(2+0!)!+(2+2)!
31 =INT((SQRT(2)+0)*22)
32 =(2^(0!+2+2))
33 = INT(SQRT(((2+0!)!)!*SQRT(2))+2)
34 =((2+0!)!^2)-2
35 = ((2+0!)!^2)-INT(SQRT(2))
36 = SQRT((2+0!)!^2^2)
37 = ((2+0!)!^2)+INT(SQRT(2))
38 =((2+0!)!^2)+2 = (20*2)-2 = INT(SQRT(2+FACT(0))*22)
39 = (20*2)-INT(SQRT(2))
40 = (20*2)/INT(SQRT(2)) = 20*SQRT(2*2)
41 = INT(SQRT(2+0!) * (2+2)!) = (20*2)+INT(SQRT(2))
42 = (20*2)+2
43 = INT(20*((SQRT(SQRT(SQRT(2))))+(SQRT(SQRT(SQRT(2))))))
44 = (2+0)*22 =[SQRT(2022)]
45 = [SQRT(SQRT((2+0)^22))]
46 = INT(SQRT(SQRT(SQRT(SQRT((((2+0)+(2+2)!)!))))))
47 = INT(20*(SQRT(SQRT(2))+SQRT(SQRT(2))))
48 =(2+0)*(2*2)!
49 = INT(SQRT(SQRT(SQRT(FACT(20))))/(2+2))
50 = INT(SQRT(SQRT(SQRT(SQRT(SQRT(FACT((2+0)*22)))))))
51 = INT( ( SQRT ( ( (2+0!) ! ) ! ) ) *2 ) -2 )
52 = INT( ( SQRT ( ( (2+0!) ! ) ! ) ) *2 ) - SQRT(2) )
53 = INT(SQRT((2+0!)!)*22)
54 = INT(SQRT(SQRT(SQRT(SQRT(SQRT(SQRT(SQRT( (INT(SQRT(SQRT(SQRT(SQRT( ((SQRT(2)+0)*22)!))))))! ))))))))
55 = INT(SQRT(((2+0!)!)!)*2)+2
56 = INT(20*2*SQRT(2))


Here are some other answers:
0 = 2 * 0 * 22
      or      2 x 0 x 2 x 2
1 = 2- 0! + 2 - 2
      or      (2*0)+(2/2)
      or      -2 - 0! + 2 + 2
2 = -20 + 22
      or      2 * 0 * 2 + 2
3 = 2 + 0! + 2 - 2
      or      (2^0)^2 + 2
      or      -2 + 0! + 2 + 2
4 = 2 * 0 + 2 * 2
      or      (2*0)+2+2
5 = 2 ^ 0 + 2 * 2
      or      20/(2+2)
      or       20 / ( 2+ 2)
      or       20 / 2^2
6 = 2 + 0 + 2 + 2
      or      2 (-0! + 2 + 2 )
7 = 2 + 0! + 2 + 2
      or      [ sqrt(202)] / 2
8 = (2 + 0) * (2 + 2)
      or      (2+0)*2*2
      or      20 / 2 - 2
      or       2 * 0! * 2 * 2
9 = (2 - 0! + 2) ^ 2
      or      ((2+0!)!/2)^2
      or       int[sqrt(20) + sqrt(22)]
10 = [ sqrt(20)] * 2 + 2
      or      (2+0!+2)*2
      or      2 (0! + 2 + 2)
11 = [ sqrt(202 / sqrt(2)) ]
      or      ((2+0!)^2)+2
      or       (20+2) / 2
12 = (2 + 0!) * (2 + 2)
      or      (20/2)+2
      or      int (sqrt(202)) - 2
13 = [sqrt(202)] - [sqrt(2)]
      or      ((2+0!)!)*2)+int(sqrt(2))
      or       int(sqrt{(sqrt 2)*(0!+2+2) !)})
14 = [sqrt(20)]! / 2 + 2
      or   ((2+0!)!*2)+2
      or   [sqrt(sqrt(sgrt(sqrt(20!))))] + 2 - 2
15 = [20 - sqrt((2 * 2)!)]
      or      int(sqrt(((2+0!)!)!))/2)+2
      or      [20 - sqrt(22)]
      or   [sqrt{ 2*(0!+2+2) !}]
      or   [sqrt(sqrt(sgrt(sqrt(20!))))] + 2 -             [sqrt(2)]
16 = 20 - 2 - 2
      or      (2 + 0 + 2)^2
17 = [20 - (sqrt(sqrt((2 + 2)!)))]
      or      20-2-int(sqrt(2))
      or       [sqrt(20) * 2 * 2]
      or      20 - 2 - [sqrt(2)]
18 = 20 - sqrt(2 * 2)
      or      ((2+0!)^2)*2
      or      (sqrt(20))^2 - 2
      or      [sqrt(20)]^2 + 2
      or       20 - [sqrt(2)] - [sqrt(2)]
19 = 20 - ((2 - 2)!)
      or      -(2+0!)+22
      or       20-2/2
      or      20 + 2 - [sqrt(2)]
20 = 20 + 2 - 2
21 = -2 + 0! + 22
      or      20+(2/2)
      or      20 + 2 - [sqrt(2)]
22 = 2 * 0 + 22
      or      (2 + 0 + 2)! - 2
23 = 2 - 0! + 22
      or      2^0 + 22
      or      20 + 2 + [sqrt(2)]
24 = 20 + 2 + 2
      or      2 + 0 + 22
25 = 2 + 0! + 22
      or      ((2+0!)+2)^2
      or      (2 + 0 + 2)! + [sqrt(2)]
26 = [20 * sqrt(2)] - 2
      or      2+0+(2+2)!
      or      (2 + 0 + 2)! + 2
27 = 2 + 0! + (2 + 2)!
      or       [sqrt((2^0 + 2)!)!] + [sqrt(2)]
28 = (2 + 0!)! + 22
      or       [sqrt(202)] * 2
29 = [sqrt(((2 + 0!)!)!)] + 2 + [sqrt(2)]
30 = (2 + 0!)! + (2 +2 )!
      or      [sqrt(((2 + 0!)!)!)] + 2 + 2
31 = int((sqrt(2)+0)*22)
32 = (2^(0!+2+2))
33 = int(sqrt(((2+0!)!)!*sqrt(2))+2)
34 = ((2+0!)!^2)-2
35 = ((2+0!)!^2)-int(sqrt(2))
36 = sqrt((2+0!)!^2^2)
37 = ((2+0!)!^2)+int(sqrt(2))
38 = ((2+0!)!^2)+2
      or       (20*2)-2
39 = (20*2)-int(sqrt(2))
40 = (20*2)/int(sqrt(2))
      or       20*sqrt(2*2)
41 = int(sqrt(2+0!) * (2+2)!)
      or       (20*2)+int(sqrt(2))
42 = (20*2)+2
43 = int(20*((sqrt(sqrt(sqrt(2))))+
          (sqrt(sqrt(sqrt(2))))))
44 = (2+0)*22
      or       [sqrt(2022)]
45 = [sqrt(sqrt((2+0)^22))]
46 = int(sqrt(sqrt(sqrt(sqrt((((2+0)+
          (2+2)!)!))))))
47 = int(20*(sqrt(sqrt(2))+sqrt(sqrt(2))))
48 = (2+0)*((2*2))!
49 = int(sqrt(sqrt(sqrt((20)!)))/4)
50 = int(sqrt(sqrt(sqrt(sqrt(sqrt(
          ((2+0)*22)!))))))




Correctly solved by:

1. Colin (Yowie) Bowey   **   (56 consecutive) Beechworth, Victoria, Australia
2. Ivy Joseph **   (28 consecutive) Pune, Maharashtra, India
3. Brijesh Dave Mumbai City, Maharashtra, India
4. Kelly Stubblefield **   (26 consecutive) Mobile, Alabama
5. Sara Gagler Central High School,
Grand Junction, Colorado
6. Ritwik Chaudhuri **   (44 consecutive) Shantiniketan, West Bengal, India
7. Milos Vukovic **   (100 consecutive) Budapest, Hungary

      ** Extra Credit


Send any comments or questions to: David Pleacher