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How can we put any mathematical sign (addition, subtraction, multiplication, or division) between five $5$s to get $66$?

$$55555\;=\;66$$

Blue
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user141930
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3 Answers3

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If only addition, subtraction, multiplication, and division are allowed (with no use of parenthesis) only the following positive integers can be formed:

3, 4, 5, 6, 7, 10, 14, 15, 16, 19, 20, 21, 25, 29, 30, 31, 40, 45, 55, 100, 115, 124, 125, 126, 135, 150, 620, 630, 3125

Note that the list does not include 66.

If concatenation is allowed (but parenthesis are still disallowed) we can get the additional positive integers:

1, 11, 12, 35, 36, 44, 49, 50, 51, 59, 60, 61, 66, 70, 75, 85, 105, 106, 116, 180, 220, 250, 265, 274, 275, 276, 285, 300, 330, 500, 530, 545, 554, 555, 556, 565, 580, 605, 610, 1111, 1370, 1380, 2770, 2780, 3020, 3030, 5550, 5560, 6875, 13875, 15125, 27775, 30525, 55555
Peter Kagey
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  • But @achillehui gets $55 + 55/5 = 66$. How did you come up with that? – k.stm Nov 22 '16 at 19:40
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    @achillehui uses concatenation, which isn't one of the four stated operations. – Peter Kagey Nov 22 '16 at 19:41
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    The OP hasn't yet answered the first question under the post, which is whether concatenation is allowed. The ? question marks were inserted in @Blue's edit, but it's not entirely clear what the OP actually meant. – dxiv Nov 22 '16 at 19:48
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    @dxiv: I have removed the ?s, which inadvertently suggested that concatenation isn't allowed. (As answers have shown, concatenation is required.) Poor symbolicating on my part. Sorry, all. – Blue Nov 22 '16 at 20:13
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As you can check with a few lines of Python, there is no way to achieve this;

ops = ['+', '-', '*', '/']
results = [eval('5' + a + '5' + b + '5' + c + '5' + d + '5')
           for (a, b, c, d) in itertools.product(ops, repeat=4)]
66 in results  # False

Ordered by size, the possible results are -620, -125, -124, -115, -100, -45, -30, -29, -24.8, -21, -20, -19, -15, -14, -10, -9.8, -6, -5, -4.96, -4, -3, -0.8, -0.2, 0.008, 0.2, 0.8, 1.2, 3, 4, 4.96, 5, 5.04, 6, 7, 9.8, 10, 10.2, 14, 15, 16, 19, 20, 21, 24.8, 25, 25.2, 29, 30, 31, 40, 45, 55, 100, 115, 124, 125, 126, 135, 150, 620, 630, and 3125.

fuglede
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$$5/5 + 5\cdot 5/5 = 6 + [6-6] / [6 \cdot 6 \cdot 6]$$

James S. Cook
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