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Question 1

Complete Table 2 with 5-digit numbers whose digits are ‘1’, ‘0’, ‘6’, ‘3’, and ‘9’ in some order. Only a coloured cell should have a number greater than all its neighbours.

The biggest number in the table is _________ . The smallest even number in the table is _________ . The smallest number greater than $50,000$ in the table is _________ . Once you have filled the table above, put commas appropriately after the thousands digit.

Solution

Understand the Question

We need to form 5-digit numbers using the digits $0, 1, 3, 6,$ and $9$ without repetition. The first digit cannot be $0$ .
Rule: A cell is greater than all its adjacent neighbours (top, bottom, left, right) if and only if it is a coloured cell. A non-coloured cell cannot be greater than all of its neighbours.
Once the table is completed, we determine:
1. The biggest number in the table.
2. The smallest even number (units digit must be $0$ or $6$ ).
3. The smallest number strictly greater than $50,000$ .

Step 1 · Find the Largest Possible Number

To form the largest 5-digit number from digits $1, 0, 6, 3, 9$ , arrange them in descending order: $9 > 6 > 3 > 1 > 0$ Diagram 1

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

Step 2 · Fill the Top-Left Coloured Cell

Cell $C(1,2)$ contains $96,301$ and is not coloured, so it cannot be greater than all its neighbours ( $C(1,1)$ , $C(1,3) = 36,109$ , and $C(2,2) = 13,609$ ).

Since $36,109 < 96,301$ and $13,609 < 96,301$ , cell $C(1,1)$ must be greater than $96,301$ .

Since $C(1,1)$ is a coloured cell, assign the largest possible number: $C(1,1) = 96,310$

Step 3 · Fill Cells Adjacent to C(4,2)

Cell $C(4,2) = 10,963$ is a coloured cell, so all its neighbours ( $C(3,2)$ , $C(4,1)$ , $C(4,3)$ ) must be smaller than $10,963$ .

Possible unused 5-digit numbers smaller than $10,963$ are $10,369$ , $10,396$ , $10,639$ , $10,693$ , and $10,936$ .

Assigning valid values:

\begin{aligned} C(3,2) &= 10,639 \\ C(4,1) &= 10,369 \\ C(4,3) &= 10,936 \end{aligned}

Step 4 · Fill the Top-Right Coloured Cell

Cell $C(1,4)$ is coloured and adjacent to $C(1,3) = 36,109$ and $C(2,4) = 19,306$ .

Therefore, $C(1,4)$ must be greater than both $36,109$ and $19,306$ .

Choosing the unused number $39,610$ : $C(1,4) = 39,610$

Step 5 · Fill the Bottom-Right Coloured Cell

Cell $C(4,4)$ is coloured and adjacent to $C(3,4)$ and $C(4,3) = 10,936$ .

Therefore, $C(4,4)$ must be greater than both neighbours.

Choosing the unused number $31,906$ : $C(4,4) = 31,906$

Step 6 · Fill Remaining Cells and Complete Table

The remaining empty cells are $C(2,1)$ , $C(3,1)$ , and $C(3,4)$ , with remaining numbers $93,610$ , $93,601$ , and $30,196$ .

Assigning:

\begin{aligned} C(2,1) &= 93,610 \\ C(3,1) &= 93,601 \\ C(3,4) &= 30,196 \end{aligned}

Completed Table:

\begin{array}{|c|c|c|c|} \hline \text{96,310 (Coloured)} & \text{96,301} & \text{36,109} & \text{39,610 (Coloured)} \\ \hline \text{93,610} & \text{13,609} & \text{60,319 (Coloured)} & \text{19,306} \\ \hline \text{93,601} & \text{10,639} & \text{60,193} & \text{30,196} \\ \hline \text{10,369} & \text{10,963 (Coloured)} & \text{10,936} & \text{31,906 (Coloured)} \\ \hline \end{array}

Answer

The biggest number in the table is 96,310.

The smallest even number in the table is 10,936.
The smallest number greater than $50,000$ in the table is 60,193.

Common Mistakes

Leading Zero Error: Writing numbers like $01,369$ , which has only $4$ digits instead of $5$ .
Neighbour Rule Violation: Making an uncoloured cell larger than all surrounding neighbours; uncoloured cells must be strictly smaller than at least one neighbour.
Even Number Misidentification: Choosing $10,369$ as the smallest number instead of checking the even units digit ( $0$ or $6$ ), where $10,936$ is the smallest even number in the table.

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