Diamond Division - Splitting up this found treasure

The Discovery

One day while on an adventure, a great treasure was found in the Tenretni Cave by an Adventure Group known as Tenehtfrus. The treasure chest had beautiful diamonds in it. It was a tremendous find!

The Adventure Group decided to donate the diamonds to a local group of fewer than 10 village girls. The girls were able to divide them equally amongst themselves.

Bright Idea

After they were divided up, Thoria, one of the girls, thought it would be better to split up the diamonds by families, rather than individual girls. Among the girls, there was two groups with two sisters ( Thoria wasn't in either group ). The rest of the girls are unrelated.

The New Count

If this re-division were to happen, that would mean the diamonds per family were five more than the diamonds per girl.

Dissent

Of course there was arguments over the re-division idea. Two sets of sister were really insistent on keeping things individually divided, because each sister would get more diamonds this way.

Frustration

Before a final decision was made, one of the girls was so frustrated, she said she didn't want ANY diamonds, she was out. Her share was then divided among the other girls - so each had more diamonds, but still all had en equal share.

Her Prerogative

Thoria changed her mind. She withdrew her suggestion of dividing by family. She was cool with the way things are now.

Final Count

How many diamonds did each girl end up with, and how many girls shared in the division?

Another solution:

Let $$G$$ be the number of girls in the village and $$d$$ be the number of Diamonds.
Then $$(G-2)$$ is the number of families,

Equation we have:

$$\frac d{G-2}-\frac dG=5$$

So

$$2d=5G(G-2)$$

As

$$2|5G(G-2)$$, so $$2|G$$, so let $$G=2G'$$. $$\begin{split}2d&=20G'(G'-1)\\d&=20\cdot\frac{G'(G'-1)}2\end{split}$$

As

$$\frac{G'(G'-1)}2$$ is an integer, $$20|d$$.
Checking values of $$d$$, we get $$(G,d)=(4,20),(6,60)$$

But after a girl left the share, the share is still equal, so

$$(G,d)=(6,60)$$ is the valid solution.

In conclusion

There are $$6-1=5$$ girls in the share with $$\frac{60}5=12$$ diamonds per girl.

• @Kwan, and Strack - it's solutions like yours that you both posted that motivates me to spend my free time posting puzzles for my kids to study and learn. Screen shots are taken and studied and discussed in group, and among fellow programming students as well. Thank you for contributing not only the answer, but a lesson as well. Apr 10, 2020 at 14:32

There are:

a total of 60 diamonds, and in the end 5 girls shared them, each having 12 diamonds.

Reasoning:

We know that the number of girls is less than 10, and that the diamonds can be equally divisible among that number, and one fewer than that (after 1 girl drops out), and two fewer than that (if divided among families instead). We also know that when the diamonds are spread equally among the families (two fewer than the girl amount), each family gets five more than the per-girl amount.
This can be satisfied if there are 60 diamonds and 6 girls at first. Spread equally, each girl gets 10 diamonds. Spread among a group of one fewer (5), each girl gets 12 diamonds. Spread among "families" (4 total), each family would get 15, which is 5 more than the original per-girl amount.

Note: I misread the problem at first, so gave the wrong answer. I used math with my original, mistaken solution, but just used trial-and-error for the second attempt.