Math Problems That Equal 10

This problem went viral when Google featured it in a Japanese commercial for Nexus 7.

Brand ten from the numbers 1, 1, 5, 8.

You can use the operations + – × ÷ ().

You take to apply all the numbers, and use each number exactly once.

The operations can be repeated (like ane + one + 5 + 8) and y'all don't take to utilise each operation.

However, y'all must use just these operations. You cannot utilise exponents so 8 + 1 + one^v = 10 is not a solution.

Tin can you effigy it out?
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"All will be well if you lot use your mind for your decisions, and listen just your decisions." Since 2007, I accept devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over i,000 free articles with no ads thanks to community support! Aid out and become early admission to posts with a pledge on Patreon.

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Reply To Viral Math Problem: Can yous make x from the numbers one, 1, 5, and viii?

This is really a pretty tricky problem! I explain the intuition in the video.

Viral Math Problem From Google – The 10 Puzzle

Beneath is a text explanation and a description of how to bank check all possibilities.

Finding the solution (text)

You accept to play around with the numbers.

Offset, notice the factors of the numbers.

10 = 2 x 5
8 = 2 x 2 ten 2
5 = 5
i = 1
1 = ane

In social club to brand ten, y'all demand to combine the 5 with the cistron of 2 from 8. Y'all somehow want to get rid of the remaining ii×2 cistron from eight. But how do you do that?

You tin can experiment with some values. Consider all the values you can get by dividing two numbers. The possible answers are:

8/5
five/8
5/1
1/5

The concluding choice 1/5 is interesting because 1 – one/5 = 4/5. So this gives you the four = 2×2 gene, and now yous combine it cleverly with the 8.

Answer

Hither's the remarkable answer to this tricky puzzle:

8/(1 – 1/5) = 10

8 ÷ (1 – 1 ÷ v) = x

The flim-flam is that i – 1/five = four/5 = 0.8, and then you take eight divided by this to go to ten.

Brute forcefulness calculation

It'due south nearly impossible that you would gauge this answer by risk!

By my calculations, this is the simply answer from ALL possibilities. Also, well-nigh every other number could exist arrived at past multiple methods. And so this is really a good puzzle: y'all're non likely to find it by simply guessing, and it's surprising once you see the result.

Here'southward how I checked every possibility. First, I considered writing the numbers 1, 1, v, 8 in a single row. How many ways are at that place?

The answer is 12. There are four! = 24 ways to arrange four elements, and so the numbers ane repeat, and so nosotros accept to divide by ii! to find there are 12 unique ways to arrange the numbers 1, ane, 5, and 8.

Hither are the 12 ways.

1, one, 5, 8
1, ane, 8, five
1, 5, i, 8
ane, 5, 8, 1
1, 8, i, five
one, 8, 5, 1
5, 1, 1, viii
5, ane, viii, 1
5, 8, 1, 1
8, 1, one, five
viii, 1, 5, 1
8, 5, ane, ane

At present I wanted to evaluate a mathematical expression using the operations +, -, ×, ÷.

I considered the expression from left to correct. The start two numbers could be evaluated with iv possible choices: +, -, ×, ÷.

For example, the pair i, 5 can atomic number 82 to 1 + 5 = 6, 1 – v = -4, i × 5 = 5, and 1 ÷ five = 0.ii.

And nosotros as well have the pair v, i that leads to v + 1 = 6, v – 1 = 4, 5 × 1 = 5, and 5 ÷ 1 = 5.

In this style I figured out the results of picking 2 numbers.

It got trickier for the tertiary number. Consider the line 1, 8, v, 1 and I added the numbers ane + 8.

If the 3rd operation was division, and so I could have (result/number) to go:

(1 + 8)/5

But I also needed to consider the flip side (number/consequence), which is the different expression:

5/(1 + viii)

So I made sure to consider both possibilities.

I did a similar thing for subtraction: when evaluating from left to correct, I evaluated both (number – result) and (result – number).

When a line had both subtraction and division, I had to consider all the combinations to make sure to enumerate all the possibilities.

Addition and multiplication are commutative so they don't have this issue [since (number + result) = (effect + number) and (number × result) = (upshot × number)].

Here's a spreadsheet of my calculations on Google Docs:

1158 puzzle all possibilities

The answer of 10 can be institute on line 1755.

It'southward amazing this is the only style to go far at 10, and it's a pretty clever puzzle!

I learned about the puzzle via NYT Wordplay http://wordplay.blogs.nytimes.com/2015/09/07/24/?_r=0

And that post credits 7 Puzzle web log http://7puzzleblog.com/209/

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