Each cluster of three numbers follows the same hidden rule to produce the middle number. Solve for the missing value from the pattern shown by the other cluster. Game: Rule Hunter — https://play.google.com/store/apps/details?id=com.makenumber

Each pair of numbers produces the number below it by the same hidden rule. Solve for the missing number from the pattern shown by the other pairs. Game: Rule Hunter — https://play.google.com/store/apps/details?id=com.makenumber

Each diamond follows the same simple rule. What is the missing number? From my free Android game Rule Hunter: https://play.google.com/store/apps/details?id=com.makenumber

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Guys, I was randomly playing with numbers and noticed something weird lol.

3 + 3/2 = 3 × 3/2

Both give 4.5. I got curious and started checking if there is any way to generalise this.

So I started with

a + b = ab

Then rearranged it:

ab - b = a

b(a - 1) = a

So,

b = a/(a - 1)

Then I realised I can take almost any number and find another number which satisfies this.

For example, if I take 8, then b = 8/7.

8 + 8/7 = 8 × 8/7 = 64/7

I also tried another approach. If I say both the sum and product should equal some number c, then

a + b = c

ab = c

Using the quadratic formula, I got

a, b = (c ± √(c² - 4c))/2

So if I choose c = 88, I get two numbers whose sum and product are both 88.

I know this is probably already known maths and not some new discovery haha, but I was happy that I managed to derive the formula myself.

Is there any particular name for this relationship, or anything interesting related to it? Just curious if this leads to something else.

Any order is allowed.
Use each die exactly once.
Parentheses are allowed.
Only +, −, × and ÷ are allowed. No concatenation or exponents.
Exact division only.
No negative numbers.
Not every operation has to be used.

How many solutions can you find?

Please use spoiler tags for solutions:
>!your solution here!<