Integermania! - Mile and a Foot

There are 5281 feet in a mile and a foot.  Create each of the positive integers using one copy of each of the digits 5, 2, 8, and 1, and any standard operations.  All four numbers must be used, but no others.  Your solutions will be assigned an exquisiteness level.

Computer analysis shows that if a set contains 4 digits, then the largest possible level 1 exquisiteness is achieved by the digits in the number 5281.  Therefore, the name "Mile and a Foot" seems very appropriate, because this set will produce a very long sequence of level 1 solutions.  This result suggested some additional questions.

Pellegrini Paolo has reported that the set {2, 3, 4, 22} produces an exquisiteness just one more than the exquisiteness of "Mile and a Foot".  This result answers both questions above, but the questions can be rephrased for this new set.  Does the set {2, 3, 4, 22} give the largest possible exquisiteness?

Use the online submissions page to get your Integermania solutions posted here!  Five "new" or "improved" solutions per person per month are accepted.

Note:  If the math symbols are not displaying correctly, you need to upgrade your browser to either IE 6.0 or Netscape 6.0 or better.  This page will print nicely using the "shrink to fit" option in Netscape, with landscape orientation and narrow margins.

  1 (1.0)
8 – 5 – 2 × 1
Steve Wilson, 9/09
Raytown, MO		 2 (1.0)
8 – 5 – 2 + 1
Steve Wilson, 9/09
Raytown, MO		 2 (1.0)

Steve Wilson, 9/09
Raytown, MO		 4 (1.0)
8 + 2 – 5 – 1
Steve Wilson, 9/09
Raytown, MO		 5 (1.0)
8 + 2 – 5 × 1
Steve Wilson, 9/09
Raytown, MO		 6 (1.0)
2 + 8 + 1 – 5
Carolyn Neptune, 9/09
Prairie Village, KS		 7 (1.0)
5 × (2 + 1) – 8
Carolyn Neptune, 9/09
Prairie Village, KS		 8 (1.0)
+ 5 – 1
Carolyn Neptune, 9/09
Prairie Village, KS		 9 (1.0)

Carolyn Neptune, 9/09
Prairie Village, KS		 10 (1.0)
8 + 5 – 2 – 1
Carolyn Neptune, 9/09
Prairie Village, KS
  11 (1.0)
8 + 5 – 2 × 1
Ralph Jeffords, 9/09
Centreville, VA		 12 (1.0)
8 + 5 – 2 + 1
Ralph Jeffords, 9/09
Centreville, VA		 13 (1.0)
(8 + 1) × 2 – 5
Ralph Jeffords, 9/09
Centreville, VA		 14 (1.0)
8 + 5 + 2 – 1
Ralph Jeffords, 9/09
Centreville, VA		 15 (1.0)
8 + 5 + 2 × 1
Ralph Jeffords, 9/09
Centreville, VA		 16 (1.0)
8 + 5 + 2 + 1
Kevin Schwarz, 10/09
Olathe, KS		 17 (1.0)
5 × 2 + 8 – 1
Kevin Schwarz, 10/09
Olathe, KS		 18 (1.0)
5 × 2 + 8 × 1
Kevin Schwarz, 10/09
Olathe, KS		 19 (1.0)
5 × 2 + 8 + 1
Kevin Schwarz, 10/09
Olathe, KS		 20 (1.0)
× 5 × 1
Kevin Schwarz, 10/09
Olathe, KS
  21 (1.0)
5 × + 1
Kevin Schwarz, 9/09
Olathe, KS		 22 (1.0)
8 × 2 + 5 + 1
Kevin Schwarz, 9/09
Olathe, KS		 23 (1.0)
2 × (8 + 1) + 5
Ralph Jeffords, 10/09
Centreville, VA		 24 (1.0)
(5 + 1) ×
Carolyn Neptune, 10/09
Prairie Village, KS		 25 (1.0)
(8 + 5) × 2 – 1
Kevin Schwarz, 9/09
Olathe, KS		 26 (1.0)
(8 + 5) × 2 × 1
Kevin Schwarz, 9/09
Olathe, KS		 27 (1.0)
(8 + 5) × 2 + 1
Kevin Schwarz, 9/09
Olathe, KS		 28 (1.0)
2 × (8 + 5 + 1)
Carolyn Neptune, 10/09
Prairie Village, KS		 29 (1.0)
8 × (2 + 1) + 5
Carolyn Neptune, 10/09
Prairie Village, KS		 30 (1.0)
(8 – 2) × 5 × 1
Carolyn Neptune, 10/09
Prairie Village, KS
  31 (1.0)
5 × (8 – 2) + 1
Ralph Jeffords, 10/09
Centreville, VA		 32 (1.0)
8 × (5 – 2 + 1)
Steve Wilson, 11/09
Raytown, MO		 33 (1.0)
5 × (8 – 1) – 2
Steve Wilson, 11/09
Raytown, MO		 34 (1.0)
(5 – 1) × 8 + 2
Carolyn Neptune, 10/09
Prairie Village, KS		 35 (1.0)
5 × (8  – 2 + 1)
Steve Wilson, 10/09
Raytown, MO		 36 (1.0)
(5 + 1) × (8  – 2)
Steve Wilson, 10/09
Raytown, MO		 37 (1.0)
5 × 8  – 2 – 1
Steve Wilson, 10/09
Raytown, MO		 38 (1.0)
5 × 8  – 2 × 1
Steve Wilson, 10/09
Raytown, MO		 39 (1.0)
5 × 8  – 2 + 1
Steve Wilson, 10/09
Raytown, MO		 40 (1.0)
5 × 8 × (2 – 1)
Di Xiao, 10/09
Ottawa, ON
  41 (1.0)
5 × 8 + 2 – 1
Di Xiao, 10/09
Ottawa, ON		 42 (1.0)
5 × 8 + 2 × 1
Di Xiao, 10/09
Ottawa, ON		 43 (1.0)
5 × 8 + 2 + 1
Di Xiao, 10/09
Ottawa, ON		 44 (1.0)

Ralph Jeffords, 10/09
Centreville, VA		 45 (1.0)
5 × (8 + 2 – 1)
Di Xiao, 10/09
Ottawa, ON		 46 (1.0)
8 × (5 + 1) – 2
Carolyn Neptune, 11/09
Prairie Village, KS		 47 (1.0)
5 × (8 + 1) + 2
Carolyn Neptune, 11/09
Prairie Village, KS		 48 (1.0)
8 × (5 + 2 – 1)
Carolyn Neptune, 11/09
Prairie Village, KS		 49 (1.0)
(5 + 2) × (8 – 1)
Carolyn Neptune, 11/09
Prairie Village, KS		 50 (1.0)
5 × (8 + 2) × 1
Carolyn Neptune, 11/09
Prairie Village, KS
  51 (1.0)
5 × (8  + 2) + 1
Steve Wilson, 11/09
Raytown, MO		 52 (2.0)
5 × 8 + 12
Pellegrini Paolo, 10/09
Martina Franca, Italy		 53 (2.2)
58 –
Pellegrini Paolo, 10/09
Martina Franca, Italy		 54 (2.0)
18 × (5 – 2)
Steve Wilson, 1/10
Raytown, MO		 55 (1.0)
(5 + 2) × 8 – 1
Megan Kell, 11/09
Overland Park, KS		 56 (1.0)
(5 + 2) × 8 × 1
Megan Kell, 11/09
Overland Park, KS		 57 (1.0)
(5 + 2) × 8 + 1
Megan Kell, 11/09
Overland Park, KS		 58 (2.0)
58 × (2 – 1)
Pellegrini Paolo, 10/09
Martina Franca, Italy		 59 (2.0)
52 + 8 – 1
Pellegrini Paolo, 10/09
Martina Franca, Italy		 60 (1.0)
(5 + 1) × (8  + 2)
Steve Wilson, 11/09
Raytown, MO
 
  61 (2.0)
5 × 8 + 21
Pellegrini Paolo, 10/09
Martina Franca, Italy		 62 (2.2)
– 8
Steve Wilson, 12/09
Raytown, MO		 63 (1.0)
(5 + 2) × (8  + 1)
Steve Wilson, 11/09
Raytown, MO		 64 (1.0)
(5 + 2 + 1) × 8
Megan Kell, 11/09
Overland Park, KS		 65 (2.0)
(21 – 8) × 5
Pellegrini Paolo, 11/09
Martina Franca, Italy		 66 (2.2)
+ 1
Pellegrini Paolo, 11/09
Martina Franca, Italy		 67 (2.0)
82 – 15
Amelia Rylko, 11/09
Lawrence, KS		 68 (2.0)
12 × 5 + 8
Kevin Schwarz, 12/09
Olathe, KS		 69 (2.4)
– 5 + 2
Ralph Jeffords, 12/09
Centreville, VA		 70 (1.0)
(8 – 1) × 5 × 2
Ralph Jeffords, 11/09
Centreville, VA
  71 (2.0)
81 – 2 × 5
Pellegrini Paolo, 11/09
Martina Franca, Italy		 72 (1.0)
(5 × 2 – 1) × 8
Ralph Jeffords, 11/09
Centreville, VA		 73 (2.0)
85 – 12
Ralph Jeffords, 12/09
Centreville, VA		 74 (2.0)
81 – 5 – 2
Ralph Jeffords, 12/09
Centreville, VA		 75 (1.0)
(8 × 2 – 1) × 5
Ralph Jeffords, 11/09
Centreville, VA		 76 (2.0)
82 – 5 – 1
Kevin Schwarz, 12/09
Olathe, KS		 77 (2.0)
82 – 5 × 1
Ralph Jeffords, 12/09
Centreville, VA		 78 (1.0)
(8 × 5 – 1) × 2
Ralph Jeffords, 11/09
Centreville, VA		 79 (1.0)
8 × 5 × 2 – 1
Ralph Jeffords, 11/09
Centreville, VA		 80 (1.0)
5 × 2 × 8 × 1
Mary Tuggle, 9/09
Kansas City, MO
  81 (1.0)
5 × 2 × 8 + 1
Mary Tuggle, 9/09
Kansas City, MO		 82 (1.0)
2 × (8 × 5 + 1)
Kevin Schwarz, 11/09
Olathe, KS		 83 (2.0)
85 – 2 × 1
Kevin Schwarz, 11/09
Olathe, KS		 84 (2.0)
85 – 2 + 1
Kevin Schwarz, 11/09
Olathe, KS		 85 (1.0)
(1 + 2 × 8) × 5
Pellegrini Paolo, 9/09
Martina Franca, Italy		 86 (2.0)
82 + 5 – 1
Kevin Schwarz, 11/09
Olathe, KS		 87 (2.0)
82 + 5 × 1
Kevin Schwarz, 11/09
Olathe, KS		 88 (1.0)
(1 + 2 × 5) × 8
Pellegrini Paolo, 9/09
Martina Franca, Italy		 89 (2.2)
5 × (18 – .2)
Steve Wilson, 12/09
Raytown, MO		 90 (1.0)
(1 + 8) × 2 × 5
Pellegrini Paolo, 9/09
Martina Franca, Italy
  91 (2.0)
12 × 8 – 5
Kevin Schwarz, 12/09
Olathe, KS		 92 (2.0)
18 × 5 + 2
Amelia Rylko, 11/09
Lawrence, KS		 93 (3.2)
5! – 28 + 1
Kenneth Chapman, 12/09
Santa Paula, CA		 94 (2.0)
51 × 2 – 8
Amelia Rylko, 11/09
Lawrence, KS		 95 (2.2)
+ 5
Steve Wilson, 12/09
Raytown, MO		 96 (1.0)
(1 + 5) × 2 × 8
Pellegrini Paolo, 9/09
Martina Franca, Italy		 97 (2.0)
21 × 5 – 8
Amelia Rylko, 11/09
Lawrence, KS		 98 (2.4)
2 × + 8
Steve Wilson, 12/09
Raytown, MO		 99 (2.4)

Carolyn Neptune, 12/09
Prairie Village, KS		 100 (2.0)
12.5 × 8
Steve Wilson, 12/09
Raytown, MO
 
  101 (2.0)
12 × 8 + 5
Kevin Schwarz, 12/09
Olathe, KS		 102 (2.0)
85 × 1.2
Steve Wilson, 2/10
Raytown, MO		 103 (2.4)
85 +
Steve Wilson, 2/10
Raytown, MO		 104 (2.0)
8 × (15 – 2)
Carolyn Neptune, 12/09
Prairie Village, KS		 105 (2.2)
+ 25
Kenneth Chapman, 12/09
Santa Paula, CA		 106 (2.0)
85 + 21
Kevin Schwarz, 12/09
Olathe, KS		 107 (3.0)
– 1
Pellegrini Paolo, 12/09
Martina Franca, Italy		 108 (2.2)
+ 58
Steve Wilson, 1/10
Raytown, MO		 109 (2.0)
21.8 × 5
Steve Wilson, 2/10
Raytown, MO		 110 (2.0)
51 × 2 + 8
Steve Wilson, 2/10
Raytown, MO
  111 (2.8)
– 2 + .5
Pellegrini Paolo, 12/09
Martina Franca, Italy		 112 (2.0)
28 × (5 – 1)
Pellegrini Paolo, 12/09
Martina Franca, Italy		 113 (2.0)
21 × 5 + 8
Pellegrini Paolo, 12/09
Martina Franca, Italy		 114 (2.0)
2 × (58 – 1)
Carolyn Neptune, 12/09
Prairie Village, KS		 115 (2.0)
58 × 2 – 1
Steve Wilson, 2/10
Raytown, MO		 116 (2.0)
58 × 2 × 1
Pellegrini Paolo, 12/09
Martina Franca, Italy		 117 (2.0)
125 – 8
Karoly Szalay, 12/09
Ottawa, ON		 118 (2.0)
15 × 8 – 2
Karoly Szalay, 12/09
Ottawa, ON		 119 (2.2)
– 1
Steve Wilson, 1/10
Raytown, MO		 120 (1.0)
(1 + 2) × 5 × 8
Pellegrini Paolo, 9/09
Martina Franca, Italy
  121 (2.2)
+ 1
Steve Wilson, 1/10
Raytown, MO		 122 (2.0)
15 × 8 + 2
Steve Wilson, 1/10
Raytown, MO		 123 (2.0)
128 – 5
Karoly Szalay, 12/09
Ottawa, ON		 124 (2.2)
5 × – 1
Pellegrini Paolo, 1/10
Martina Franca, Italy		 125 (2.2)

Pellegrini Paolo, 1/10
Martina Franca, Italy		 126 (2.0)
18 × (5 + 2)
Karoly Szalay, 12/09
Ottawa, ON		 127 (2.2)

Pellegrini Paolo, 1/10
Martina Franca, Italy		 128 (2.0)
(21 – 5) × 8
Pellegrini Paolo, 1/10
Martina Franca, Italy		 129 (2.6)
+
Pellegrini Paolo, 1/10
Martina Franca, Italy		 130 (2.4)

Carolyn Neptune, 1/10
Prairie Village, KS
  131 (3.2)
5! + 8 + 2 + 1
John Kendall, 1/10
Lenexa, KS		 132 (2.2)
+ 2
Carolyn Neptune, 1/10
Prairie Village, KS		 133 (2.0)
125 + 8
Carolyn Neptune, 1/10
Prairie Village, KS		   135 (2.0)
5 × (28 – 1)
Carolyn Neptune, 1/10
Prairie Village, KS		 136 (2.0)
8 × (15 + 2)
Carolyn Neptune, 1/10
Prairie Village, KS		   138 (2.2)
– 2
Ralph Jeffords, 2/10
Centreville, VA		 139 (2.0)
28 × 5 – 1
Megan Kell, 12/09
Overland Park, KS		 140 (2.0)
28 × 5 × 1
Megan Kell, 12/09
Overland Park, KS
  141 (2.0)
28 × 5 + 1
Megan Kell, 12/09
Overland Park, KS		 142 (2.2)
+ 2
Kevin Schwarz, 1/10
Olathe, KS		   144 (2.0)
152 – 8
Ralph Jeffords, 2/10
Centreville, VA		 145 (2.0)
5 × (28 + 1)
Carolyn Neptune, 2/10
Prairie Village, KS		 146 (2.6)
+ 2
Ralph Jeffords, 2/10
Centreville, VA		 147 (3.2)
5! + 28 – 1
John Kendall, 1/10
Lenexa, KS		 148 (2.8)

Ralph Jeffords, 2/10
Centreville, VA		 149 (2.4)
8 × + 5
Ralph Jeffords, 2/10
Centreville, VA		 150 (2.0)
15 × (8 + 2)
Kevin Schwarz, 1/10
Olathe, KS
          155 (2.2)
+ 5
Pellegrini Paolo, 2/10
Martina Franca, Italy		 156 (2.0)
158 – 2
Pellegrini Paolo, 2/10
Martina Franca, Italy		 157 (2.0)
81 × 2 – 5
Pellegrini Paolo, 2/10
Martina Franca, Italy		 158 (2.2)

Pellegrini Paolo, 2/10
Martina Franca, Italy		 159 (2.2)
+ 1 – 2
Pellegrini Paolo, 2/10
Martina Franca, Italy		 160 (2.2)
(5 – 1) ×
Carolyn Neptune, 2/10
Prairie Village, KS
    162 (2.2)

Carolyn Neptune, 2/10
Prairie Village, KS		 163 (2.0)
21 × 8 – 5
Ralph Jeffords, 3/10
Centreville, VA		 164 (2.2)
82 ×
Carolyn Neptune, 3/10
Prairie Village, KS		 165 (2.2)
2 × + 5
Ralph Jeffords, 3/10
Centreville, VA		 166 (2.2)

Kevin Schwarz, 2/10
Olathe, KS		   168 (3.4)
× 1
John Kendall, 1/10
Lenexa, KS		 169 (3.0)
(5 + 8)2 × 1
Carolyn Neptune, 2/10
Prairie Village, KS		 170 (2.2)

Carolyn Neptune, 2/10
Prairie Village, KS
      173 (2.0)
21 × 8 + 5
Ralph Jeffords, 3/10
Centreville, VA		   175 (2.0)
25 × (8 – 1)
Carolyn Neptune, 3/10
Prairie Village, KS		 176 (2.8)
(5 – ) ×
Ralph Jeffords, 3/10
Centreville, VA		   178 (2.2)
– 2
Kevin Schwarz, 2/10
Olathe, KS		 179 (2.4)
5 × – 1
Ralph Jeffords, 3/10
Centreville, VA		 180 (2.0)
18 × 5 × 2
Megan Kell, 12/09
Overland Park, KS
    182 (2.2)
+ 2
Kevin Schwarz, 2/10
Olathe, KS		                
    192 (2.0)
8 × (25 – 1)
Kevin Schwarz, 2/10
Olathe, KS		             199 (2.0)
25 × 8 – 1
Kevin Schwarz, 1/10
Olathe, KS		 200 (2.0)
25 × 8 × 1
Kevin Schwarz, 1/10
Olathe, KS
  201 (2.0)
25 × 8 + 1
Kevin Schwarz, 1/10
Olathe, KS		             208 (2.0)
8 × (25 + 1)
Kevin Schwarz, 2/10
Olathe, KS		    
      233 (3.0)
152 + 8
Tyler Cox, 9/09
Olathe, KS		              
              257 (2.0)
258 – 1
Mary Tuggle, 11/09
Kansas City, MO		      
    262 (3.0)
28 + 5 + 1
Stephen Inglin, 9/09
DeSoto, KS		                
              417 (2.0)
52 × 8 + 1
Megan Kell, 11/09
Overland Park, KS		      
              527 (2.0)
528 – 1
Mary Tuggle, 11/09
Kansas City, MO		      
  581 (2.0)
582 – 1
Mary Tuggle, 11/09
Kansas City, MO		                  
                    640 (2.0)
128 × 5
Sean Collins, 1/10
Overland Park, KS
        824 (2.0)
825 – 1
Mary Tuggle, 11/09
Kansas City, MO		            
  851 (2.0)
852 – 1
Mary Tuggle, 11/09
Kansas City, MO		                  
    972 (3.2)
5! × 8 + 12
Sean Collins, 1/10
Overland Park, KS		                
                    1000 (2.0)
125 × 8
Sean Collins, 1/10
Overland Park, KS
            1216 (2.0)
152 × 8
Sean Collins, 1/10
Overland Park, KS		        
                    1720 (2.0)
215 × 8
Sean Collins, 2/10
Overland Park, KS
                2008 (2.0)
251 × 8
Sean Collins, 2/10
Overland Park, KS		    
                    9600 (3.2)
× 5
Sean Collins, 2/10
Overland Park, KS
                    100000 (3.0)
(8 + 2 × 1)5
Mary Tuggle, 9/09
Kansas City, MO
  100001 (3.0)
(2 + 8)5 + 1
Stephen Inglin, 9/09
DeSoto, KS		                  
  161051 (3.0)
(8 + 2 + 1)5
Mary Tuggle, 9/09
Kansas City, MO		                  
            16777216 (3.0)
(5 + 2 + 1)8
Stephen Inglin, 9/09
DeSoto, KS		        
                  17210369 (3.0)
285 + 1
Tyler Cox, 9/09
Olathe, KS		  
              25401607 (3.4)
7! × 7! + 8 – 1
Jordan Cordry, 2/10
Overland Park, KS		   25401609 (3.4)
7! × 7! + 8 + 1
Jordan Cordry, 2/10
Overland Park, KS		  
                  1024192511999 (3.6)
8! × 7! × 7! – 1
Jordan Cordry, 2/10
Overland Park, KS		 1024192512000 (3.6)
8! × 7! × 7! × 1
Jordan Cordry, 2/10
Overland Park, KS
  1024192512001 (3.6)
8! × 7! × 7! + 1
Jordan Cordry, 2/10
Overland Park, KS		                  
                    googol (3.6)

Ralph Jeffords, 10/09
Centreville, VA
                    googolplex (3.8)

Ralph Jeffords, 10/09
Centreville, VA