Seven Sevens to Seventy

Algebra Level 3

Using only multiplication, division, addition, subtraction, and parentheses (and no logarithms or exponents or any other kind of operation), is it possible to express each of the first seventy positive integers with exactly seven sevens? (You may use two sevens to make 77 but otherwise concatenation is not allowed.)

Yes No

1 solution

David Vreken
Jul 31, 2019

One strategy is to first express the number's closest multiple of seven with the least sevens, then add or subtract the difference with the least sevens, and use the remaining sevens to add zero (or put them in the expression without changing the value). For example, for $16$ , the closest multiple of seven is $14$ , which is $7 + 7$ , the difference between $16$ and $14$ is $2$ , which is $\frac{7 + 7}{7}$ , which leaves two sevens leftover, which can make zero by $7 - 7$ , so that $16 = 7 + 7 + \frac{7 + 7}{7} + (7 - 7)$ .

Unfortunately the numbers $25$ , $31$ , $32$ , $38$ , $60$ , and $66$ need more than seven sevens for this strategy, but using a little bit of creativity a result can still be found.

Here is a list of the first $100$ positive integers expressed with seven sevens:

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

The smallest positive integer that I could not express with seven sevens was $102$ . Please comment below if you can do it, or prove that it can't be done!

$102 = 77 + 7 + 7 + \dfrac{77}7$ .

Pi Han Goh - 1 year, 10 months ago

Great! Okay, here's the next $30$ , with the blank ones more that I couldn't find:

101	$7(7+7) + \frac{7 + 7 + 7}{7}$	111	$7(7 + 7) + 7 + 7 - \frac{7}{7}$	121	$\frac{777 + 77 - 7}{7}$
102	$77 + 7 + 7 + \frac{77}{7}$	112	$7(7 + 7) + 7 + 7 - (7 - 7)$	122	$\frac{777}{7} + \frac{77}{7}$
103	$7(7 + 7) + 7 - \frac{7 + 7}{7}$	113	$7(7 + 7) + 7 + 7 + \frac{7}{7}$	123	$\frac{777 + 77 + 7}{7}$
104	$7(7 + 7) + \frac{7 \cdot 7 - 7}{7}$	114	$\frac{777 + 7 + 7 + 7}{7}$	124	$77 + 7 \cdot 7 - \frac{7 + 7}{7}$
105	$7(7 + 7) + 7 + (7 - 7) \cdot 7$	115	$\frac{777 + 77}{7} - 7$	125	$7(\frac{77}{7} + 7) - \frac{7}{7}$
106	$7(7 + 7) + \frac{7 \cdot 7 + 7}{7}$	116	$\frac{777 - 7 - 7}{7} + 7$	126	$7(7 + 7) + 7 + 7 + 7 + 7$
107	$7(7 + 7) + 7 + \frac{7 + 7}{7}$	117	$\frac{777}{7} + 7 - \frac{7}{7}$	127	$7(\frac{77}{7} + 7) + \frac{7}{7}$
108	$(7 - \frac{7}{7})(\frac{77}{7} + 7)$	118	$\frac{777}{7} + 7 + (7 - 7)$	128	$\frac{77}{7} \cdot \frac{77}{7} + 7$
109	$\frac{777}{7} - \frac{7 + 7}{7}$	119	$7(7 + 7 + \frac{7 + 7 + 7}{7})$	129	$\frac{777 + 77}{7} + 7$
110	$\frac{77}{7}(\frac{77 - 7}{7})$	120	$\frac{777 + 7 + 7}{7} + 7$	130

Can anyone find any of these blanks?

David Vreken - 1 year, 10 months ago

$125=7(\frac{77}{7}+7)-\frac{7}{7}$

Sam Zhou - 1 year, 10 months ago

Similarly, $127=7(\frac{77}{7}+7)+\frac{7}{7}$

Sam Zhou - 1 year, 10 months ago

Also, your expression for 117 is incorrect. You probably meant $117=\frac{777}{7}+7-\frac{7}{7}$ .

Sam Zhou - 1 year, 10 months ago

Thanks @Sam Zhou ! I updated the chart.

David Vreken - 1 year, 10 months ago

I also found $124 = 77 + 7 \cdot 7 - \frac{7 + 7}{7}$ and $115 = \frac{777 + 77}{7} - 7$ .

David Vreken - 1 year, 10 months ago

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Yuriy Kazakov - 1 year, 10 months ago

Seven Sevens to Seventy

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