Answer to Question #182382 in Combinatorics | Number Theory for Niya

Question

The six digits of my employee reference code at work are all different the first digit is one and the last one is eight when the coat is written as 32 digit numbers the middle number is a square number and is exactly halfway between the other two numbers what are the middle two digits of my employee reference code

Answer 1

The six digit number

1_ _ _ _ 8

We have given that middle number is square number and exactly halfway between two

We use hit and trial method

Case -1

Middle no. = 4

Other number can be 3 or 5

Or other number can be 1 and 6

Case 2

Middle no. = 9

Other number can be 8 and 10

7, 11

6,12

5,13

4,14

3,15

2,16

1,17

Case 3

Middle no. = 16

Other number can be 15and 17

14,18

13,19

12,20

11,21

10,22 these are not possible

9, 23

8,24

7,25

6,26

5,27

4,28

3,28

2,30

1,31 these are possible

Case 4

Middle no. = 25

Other number can be 24 and 26

...

....

Possible are

9,41

..

..

1,49

Similarly further

These cases are not possible , as we have to 4 vacant places with given condition

So only case 1 is possible

the number can be

130458

or 110468

so the answer is 04 middle number of the code .