Problems & Puzzles: Puzzles

 Puzzle 928. Surreal integers Fausto Morales sent the following nice puzzle. A surreal number is a positive integer resulting from collating the numbers of an equivalent arithmetical expression, in order of appearance, using common auxiliary arithmetical symbols and operators. Examples: 713 = -7•1 + (3!)! 715 = (7-1)! - 5 Q1: Can you find more surreal numbers? A surreal prime is a prime surreal number, such as: 127 = -1 + 2^7 Q2: Can you find more surreal primes?

Contributions came from Simon Cavegn

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Simon wrote on Oct 20, 2018:

Here a few solutions I found for puzzle 928 Q2:

733=7+3!!+3!
2161=(2+1)*6!+1
4337=(4+3!!)*3!-7
12967=1*2*9*6!+7
15643=1*5^6+4!-3!
49681=(4+9)*6!+8!+1
156253=1*5^6*2*5+3 Three consecutive primes starting here.
156257=1*5^6*2*5+7
156259=1*5^6*2*5+9
161281=(1^6!+1+2)*8!+1 There are at least 128 similar solutions to this number due to +-1^(even n)=1, 1!=1, 2!=2, ...
282239=-2*8!-2-2+3+9!
287159=-2*8!+7!-1-5!+9!
343193=(-3-4!)^3-1+9!-3
354961=-3*5!*4!+9!+6!+1
357839=-3*5-7!+8+3!+9!
357977=3+5!-7!+9!+7+7
360593=-3*6!-0!-5!+9!-3!
362897=(3+6)*2-8+9!+7
362903=3*6+2+9!+0+3
362759=(-3-6)*2*7+5+9!
362903=(3*6+2+9!)*0!+3
362759=(-3+6)*2!-7-5!+9!
362867=((-3+6)^2!)*8!-6-7
362759=(3-6+2)^7-5!+9!
362941=3*6*2+9!+4!+1
362759=(3-6+2!)^7-5!+9!
362953=(3+6)^2!+9!-5-3
363619=3^6+3+6+1+9!
363659=3*6*3+6!+5+9!
363659=(3+6)*3!+6!+5+9!
364291=(3^6-4!)*2+9!+1
365039=3*6!-5+0!+3+9!
365159=3*6*5!-1+5!+9!
366479=3-6!-6!-4+7!+9!
367189=(-3-6!+7!)*1-8+9!
367259=((-3+6)^7)*2+5+9!
367909=(-3+6!)*7+9!+0!+9
368609=(-3+6!)*8-6-0!+9!
369739=(-3+6+9+7)^3+9!
372941=(3+7!)*2+9!-4!-1
393541=(3^9-3!)*5*4+1
483829=(-4+8)*3*8!-2-9
483827=4*8!*3-8+2-7
483839=(-4+8!)*3+8+3+9!
488879=(4+8)*8!+8+7!-9
488897=(4+8)*8!+8+9+7!
730799=-7+3!+0+7!+9!+9!
968041=(9+6+8!+0)*4!+1
1048309=(1+0!+4!)*8!-3+0!-9
1244167=(1+2)*4!*4!*1*6!+7
1255687=(1+2)*5!*5*6!-8!+7

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Emmanuel Vantieghem wrote on Oct 25, 2018:

Here are some surreal numbers :
145 = 1! + 4! + 5!
2592 = (2^5)*(9^2)
5167 = 5! + 1 + 6 + 7!   (prime)
5177 = 5! + 17 + 7!
5637 = -5! + 6! - 3 + 7!
6476 = 6!  - 4 +7! + 1
10077 = -1 - 0!  - 0! + 7! + 7!
40318 = 4^0 - 3^1 + 8!
40585 = 4! + 0! + 5! + 8! + 5!
80518 = 8! - 0! - 5! - 1 + 8!
117650 = 1^1 +(7^6)*(5^0) = (1^1)*(7^6) + 5^0
163855 = ((1^6)*3 + 8^5)*5
236764 = 2(3^6+7^6+4)
317489 = -3! - 1 - 7! - 4! - 8! +  9!  (prime)
357592 = -3! - 5! - 7! - 5! + 9! - 2!
357941 = 3! + 5! - 7! + 9! - 4! - 1
357966 = 3! + 5! - 7! + 9! - 6! + 6! = 3! + 5! - 7! + 9! + 6! - 6!
361469 = 3! - 6! - 1 + 4! -6! + 9!  (prime)
367209 = 3! - 6! + 7! + 2! + 0! + 9!
398173 = 3! + 9! + 8! + 1 - 7! + 3!
715799 = -7! - 1 + 5! - 7! + 9! + 9!
720599 = -7! - 2! + 0! - 5! + 9! + 9!

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Fausto wrote on Oct 26, 2018

Q1:    343 = (3 + 4) ^ 3
Q2:    131071 =  (-1 + 3) ^ (10 + 7) - 1

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