Question 1

We will fill the table using the given rules.

Step 1 — Find the largest possible number

We have digits '1', '0', '6', '3', '9'. We need to make 5-digit numbers. The first digit cannot be '0'. To make the largest number, we arrange digits from biggest to smallest. The largest digit is '9'. The next largest is '6'. Then '3', '1', '0'. So, the largest number we can make is 96,310.

$\boxed{\text{Largest number} = 96,310}$

Step 2 — Fill the top-left coloured cell

Look at cell C(1,2). It has the number 96,301. This cell is not coloured. The rule says non-coloured cells are not greater than all their neighbours. Its neighbours are C(1,1), C(1,3) (36,109), C(2,2) (13,609). Both 36,109 and 13,609 are smaller than 96,301. So, C(1,1) must be greater than 96,301. C(1,1) is a coloured cell. It must be the largest possible number, 96,310. This number is greater than 96,301. So, C(1,1) is 96,310.

$\boxed{\text{C(1,1)} = 96,310}$

Step 3 — Fill cells around C(4,2)

Cell C(4,2) has the number 10,963. This cell is coloured. The rule says coloured cells must be greater than all their neighbours. Its neighbours are C(3,2), C(4,1), C(4,3). These three cells must have numbers smaller than 10,963. The 5-digit numbers we can make that are smaller than 10,963 are: 10,369, 10,396, 10,639, 10,693, 10,936. We need to pick three of these numbers. Let us choose 10,639 for C(3,2). Let us choose 10,369 for C(4,1). Let us choose 10,936 for C(4,3). These choices satisfy the rule.

$\boxed{\text{C(3,2)} = 10,639, \text{C(4,1)} = 10,369, \text{C(4,3)} = 10,936}$

Step 4 — Fill the top-right coloured cell

Cell C(1,4) is coloured. Its neighbours are C(1,3) (36,109) and C(2,4) (19,306). So, C(1,4) must be greater than 36,109 and 19,306. We need an unused number that fits this rule. The number 39,610 is unused. 39,610 is greater than 36,109. 39,610 is greater than 19,306. So, C(1,4) is 39,610.

$\boxed{\text{C(1,4)} = 39,610}$

Step 5 — Fill the bottom-right coloured cell

Cell C(4,4) is coloured. Its neighbours are C(3,4) and C(4,3) (10,936). So, C(4,4) must be greater than C(3,4) and 10,936. We need an unused number that fits this rule. The number 31,906 is unused. 31,906 is greater than 10,936. So, C(4,4) is 31,906.

$\boxed{\text{C(4,4)} = 31,906}$

Step 6 — Fill the remaining cells

The remaining empty cells are C(2,1), C(3,1), C(3,4). The remaining unused numbers are 93,610, 93,601, 30,196. Consider C(3,1). It is not coloured. Its neighbours are C(2,1), C(3,2) (10,639), C(4,1) (10,369). C(3,1) cannot be greater than all its neighbours. If C(3,1) is 93,610 or 93,601, then C(2,1) must be larger. Let us choose C(2,1) = 93,610. Let us choose C(3,1) = 93,601. C(2,1) is a neighbour of C(1,1) (96,310). 96,310 is greater than 93,610. This is good. C(3,1) is a neighbour of C(2,1) (93,610). 93,610 is greater than 93,601. This is good. This leaves C(3,4) = 30,196. C(3,4) is a neighbour of C(3,3) (60,193) and C(4,4) (31,906). 60,193 is greater than 30,196. 31,906 is greater than 30,196. This is good. All conditions are met.

$\boxed{\text{C(2,1)} = 93,610, \text{C(3,1)} = 93,601, \text{C(3,4)} = 30,196}$

Completed Table: Row 1: 96,310 (Coloured) | 96,301 | 36,109 | 39,610 (Coloured) Row 2: 93,610 | 13,609 | 60,319 (Coloured) | 19,306 Row 3: 93,601 | 10,639 | 60,193 | 30,196 Row 4: 10,369 | 10,963 (Coloured) | 10,936 | 31,906 (Coloured)

Answer

(i) The biggest number in the table is 96,310. (ii) The smallest even number in the table is 10,936. (iii) The smallest number greater than 50,000 in the table is 60,193.