## 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.

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. I blog at learnfunfacts.com. You can find me on Twitter @EdmarkMLaw and Facebook. My email is learnfunfacts@gmail.com
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### 16 Responses to Can Eight 8’s Produce 1000?

1. Good grief! I can do that many computations. I can draw the 8 or write a poem about the 8, but I’ll never find a solution to the 8’s math puzzle. Except maybe MAYBE I could have come up with the first solution if I had really tried and didn’t mind smoke coming out of my ears.

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• There is a couple of easy solutions. I’m sure that you’d be able to find them. The other solutions are quite hard though…

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2. Garfield Hug says:

🤔My mind has been blown🙈

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3. ((8×8)+(8×8))x8-8-8-8=1000 😀

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• Correct 🙂

Thanks for participating 🙂

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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!

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5. WOw ! .. by the way thanks for stopping by my blog 🙂

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6. Nicely done!

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

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• I could do that, though who knows how long it would take.

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8. jfrels says:

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

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9. jfrels says:

Also 1000 in binary = 8 decimal

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10. printz007 says:

888+88+8+8+8 = 1000

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11. Smitha V says:

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

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