4,5,16,29
The 5 numbers = 4,5,15,16,29

Lets call the numbers ① ② ③ ④ ⑤ , in ascending order.


Average of first 3 numbers is 8.

So total of the first 3 numbers ,① + ② + ③
= 8 x 3
= 24


Average of middle 3 numbers is 12.

So total of the middle 3 numbers ,② + ③ + ④
= 12 x 3
= 36


Average of last 3 numbers is 20

So total of the last 3 numbers ,③ + ④ + ⑤
= 20 x 3
= 60

If you compare the first two, we see that ② and ③ are common.


36 - 24 = 12


This means that ④ is 12 bigger than ①.


Comparing the next two, we see that ③ and ④ are common.


60 - 36 = 24


This means that ⑤ is 24 bigger than ②.

Next,

① + ②
= 24 - ③
= 24 - 15
= 9

② + ④
= 36 - ③
= 36 - 15
= 21

④ + ⑤
= 60 - ③
= 60 -15
= 45

Lastly ,

Since ① + ② = 9, and ② is bigger than ①, we can only have the following possibilities for ① and ② :


1,8
2,7
3,6
4,5

But since ④ is bigger than ③, ④ is bigger than 15.

Knowing that ④ is 12 bigger than ①, then ① can only be 4 such that 4 + 12 = 16

The other possibilities will result in 13,14,15, which do not satisfy the above.



So ① → 4, ② → 5

Then ④ → 16 (see above)

⑤ →5 + 24 = 29