Integermania - Mile and a Foot

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.

Is the largest possible exquisiteness for a set of 4 values obtained when using the values 5, 2, 8, and 1? Or are there other values (integers larger than 9, or non-integers) which can produce a larger exquisiteness for a set of 4 values?
Is the longest possible sequence of level 1 solutions for a set of 4 values produced by the values 5, 2, 8, and 1? Or are there other values which can produce a longer sequence of level 1 solutions (possibly where the sequence does not begin with 1)?

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.

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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 (3.0) (8 – 1)² + 5 Mary Tuggle, 9/09 Kansas City, 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		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			67 (2.0) 82 – 15 Amelia Rylko, 11/09 Lawrence, KS	68 (3.0) 8² + 5 – 1 Amelia Rylko, 11/09 Lawrence, KS		70 (1.0) (8 – 1) × 5 × 2 Ralph Jeffords, 11/09 Centreville, VA
	72 (1.0) (5 × 2 – 1) × 8 Ralph Jeffords, 11/09 Centreville, VA			75 (1.0) (8 × 2 – 1) × 5 Ralph Jeffords, 11/09 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				85 (1.0) (1 + 2 × 8) × 5 Pellegrini Paolo, 9/09 Martina Franca, Italy			88 (1.0) (1 + 2 × 5) × 8 Pellegrini Paolo, 9/09 Martina Franca, Italy		90 (1.0) (1 + 8) × 2 × 5 Pellegrini Paolo, 9/09 Martina Franca, Italy
					96 (1.0) (1 + 5) × 2 × 8 Pellegrini Paolo, 9/09 Martina Franca, Italy
				115 (3.0) 8² + 51 Amelia Rylko, 9/09 Lawrence, Kansas					120 (1.0) (1 + 2) × 5 × 8 Pellegrini Paolo, 9/09 Martina Franca, Italy
		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
			824 (2.0) 825 – 1 Mary Tuggle, 11/09 Kansas City, MO
851 (2.0) 852 – 1 Mary Tuggle, 11/09 Kansas City, MO
									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
									googol (3.6) Ralph Jeffords, 10/09 Centreville, VA
									googolplex (3.8) Ralph Jeffords, 10/09 Centreville, VA