# Can Eight 8’s Produce 1000?

My previous post “Can Two 2’s Produce 5?” showed a way to produce 5 using only two 2’s and any mathematical operations. Now, here’s another similar puzzle. Using eight 8’s and any mathematical operations and symbols, can you make them equal to 1000? There are several solutions.

## Solutions

When I asked this question to some of my friends, they were able to give me six basic solutions (though there are more):

$888+88+8+8+8=1000$

$\frac{8+8}{8}(8\times 8\times 8-8)-8=1000$

$8(8\times 8+8\times 8)-8-8-8=1000$

$\frac{8888-888}{8}=1000$

$\frac{8888}{8.888}=1000$ (This solution has several variantions. Can you find them?)

$8\left [8(8+8)-\frac{8+8+8}{8} \right ]=1000$

One of them was smart enough to use square roots and factorials, so he was able to find four more solutions:

$\frac{8!}{8}-8\left [ (8\times 8\times 8)-8 \right ]-8=1000$

$8!\left [\frac{8+8}{8(88-8)} \right ]-8=1000$

$(8+8)(8\times 8)-(\sqrt{8+8})!+8-8=1000$

$\frac{8!}{8+8+8+8+8}-\sqrt{8}\times \sqrt{8}$

I have found more solutions which involve square roots and factorials. Can you find more?

This puzzle can be solved by producing 10³ using the 8’s:

$\left (\frac{88-8}{8} \right )^{(8+8+8)/8}=10^3=1000$

$\left (8+\frac{8+8}{8} \right )^{(8+8+8)/8}=10^3=1000$

If special mathematical symbols are employed, more solutions can be found:

Floor and Ceiling Functions:

$\left (\left \lfloor\sqrt{8} \right \rfloor+\left \lceil\sqrt{8} \right \rceil \right )\times\left (\left \lfloor\sqrt{8} \right \rfloor+\left \lceil\sqrt{8} \right \rceil \right )\times\left (\left \lfloor\sqrt{8} \right \rfloor+\left \lceil\sqrt{8} \right \rceil \right )\times \sqrt{8}\times \sqrt{8}=1000$

Base 8 logarithm:

$(\log_{8}8\times 8)(8\times8\times8-8)-8=1000$

Binomial Coefficient:

$\left [\binom{8+8}{\frac{8}{.8}}-8 \right ]\div 8+8-8=1000$

Gamma Function:

$\frac{8\Gamma (8)}{8+8+8+8+8}-8=1000$

$\Gamma (8)-(8\times 8)\left (8\times 8-\frac{8}{8} \right )-8=1000$

Finally, this solution involves base 8 numbers:

$\left (\frac{88}{88}\times 8\times 8\times 8 \right )_8=1000_8$

However, this is not an acceptable solution since the answer is in base 8 and 1000 in base 8 is equal to 512 in base 10. Any interested reader may try to find a valid solution using base 8 numbers.

### Posted by Edmark M. Law

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter` and Facebook. My email is edmarklaw@learnfunfacts.com

## 16 thoughts on “Can Eight 8’s Produce 1000?”

1. This is so interesting. Am going to share it with my daughter who loves Maths. Thanks for the post. Got me thinking.

Liked by 1 person

2. Do square roots count as that as an exponent of 0.5?

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3. I have £8 left in my bank account, if you could make that into £1000 I’d be much obliged.

Liked by 2 people

4. superwifeandmummy says:

You won’t believe me when I say, my next poem I’m publishing this week is about Maths. Unfortunately, I don’t think I quite share your love and fascination of it. .but fab post!

Liked by 1 person

1. Correct :)