**The ternary tree solution**

Divide the 27 balls into three groups. Put two of them on the balance. If one of the sides is heavier, the group on that side has the heavier ball. If not, the group not on the balance does. We have now localized the heavier ball to a group containing \frac{27}{\ 3\ }\ =\ 9\ balls.

Now repeat this procedure, dividing the 9 balls into groups of 3 and so on. In each weighing, we localize from 3^{m } balls to 3^{(m-1)} . It's elementary to see that we can localize to a group of 1 in 3 weighings.

It's now elementary to generalize to 3^{n}. Evidently, n weighings reduces things down to a group of 1.

**The multiple grouping solution**

Take the 27 balls and arrange them in a 3x3x3 cube. Remember this arrangement. Let's call the the axes x,y and z

First, we group the balls by planes in the x axis - you have 3 groups, each of 9 balls. Weigh two of those groups. If one is heavier, that contains the heavier ball, else the group not on the balance does Now we've isolated the heavier group of 9.

Next, we group the balls by planes on the y axis. Weigh them, now we've isolated another heavier group of 9. Remember this grouping too.

Next, we group the balls by planes on the z axis. Weigh them, now we've isolated a third heavier group of 9. Remember this grouping too.

Only one ball appears in all of the three "heavy" groupings (why?). That's your heavy ball!

How do you generalize to 3^{n} ?

I found an answer from www.quora.com

**You** are given **27 balls**, **one** of **which** is **heavier** than the others ...

... weigh **the same**. How can you **find** the **heavy ball** in 3 **weighings**? ... Now **you**
**will** be having 9 **balls with** one of them being the **heavy ball**. Divide it into 3 ...

For more information, see **You** are given **27 balls**, **one** of **which** is **heavier** than the others ...

I found an answer from www.quora.com

**You have** eight **balls** all of **the same** size. 7 of them weigh **the same** ...

First **weighing**: Weigh **three balls** against **three** other **balls** on the **balance**. ... How
**can you find** the **ball** that is **heavier** by **using** a **balance** and only two **weighings**?
.... similar coins **out** of **which one** coin is **heavy** in weight and rest are of same
weight, and **you** .... If the questioner told me, I **could verify it with** only **one**
**weighing**.

For more information, see **You have** eight **balls** all of **the same** size. 7 of them weigh **the same** ...

I found an answer from puzzling.stackexchange.com

strategy - Twelve **balls** and a **scale** - Puzzling Stack Exchange

If they are **equal**, then C3 is the odd **ball** and **it** is **heavier**. ... Because this case
implies **one** set of **balls** is **heavier**, by virtue of finding the odd **ball out**, **we** know ...
**With** each **weighing**, the **scale can** either tip to the left, tip to the right, ... There are
**27** possible ways to place each **ball** in all **three weighings**, each ...

For more information, see strategy - Twelve **balls** and a **scale** - Puzzling Stack Exchange

I found an answer from en.wikipedia.org

**Balance** puzzle - Wikipedia

A **balance** puzzle or **weighing** puzzle is a logic puzzle about **balancing** itemsâ€”
often coinsâ€”to ... In general, **with** n weighs, **you can determine** the identity of a
coin if **you have** 3^{n} ... it would take only **three weighings** to **find** the odd light coin
among **27** coins, and ... If the **balance** tips then the **heavier** coin is the odd one
**out**.

For more information, see **Balance** puzzle - Wikipedia

I found an answer from www.giss.nasa.gov

Lab 1 - Measuring Density

For example, it is possible for two pieces of metal to be made **out** of **the same**
material ... Activity 1: (**Determine** the density of an object **with** a regular shape.) ...
Measure the mass of your samples **using** the **triple beam balance** provided. **You**
**will** make this measurement **three** times and calculate the average. ... **Show** all
work.

For more information, see Lab 1 - Measuring Density

I found an answer from en.wikipedia.org

Cavendish experiment - Wikipedia

Since the gravitational force of the Earth on the small **ball could** be measured
directly by **weighing it**, the ratio of the two forces allowed the density of the Earth
to ...

For more information, see Cavendish experiment - Wikipedia

I found an answer from www.britannica.com

Energy conversion | technology | Britannica.com

Energy can either be associated **with** a material body, as in a coiled spring or a
moving object, or **it can** be independent of matter, as light and other ...

For more information, see Energy conversion | technology | Britannica.com

I found an answer from apod.nasa.gov

Astronomy Picture of the Day Archive

2019 January **27**: From the Northern to the Southern Cross .... 2018 October 07:
The **Scale** of the Universe Interactive .... 2018 April 01: I Brought **You** the Moon ...
2018 January 14: **Three** Galaxies and a Comet .... 2017 July 25: Int **Ball** Drone
Activated on the Space Station .... 2017 March 29: Nebula **with** Laser **Beams**

For more information, see Astronomy Picture of the Day Archive