The dice-and-numbers game

In maths lessons in secondary school, on certain occasions my teacher entertained our class with the following game. Each player starts with an empty 2 × 5 grid, as shown in Figure 1(a). In each round of the game, a die is rolled, and each player has to choose one of the empty boxes in their grid and fill it with the number rolled. For example, after 3 rounds, a player's grid may look like the one in Figure 1(b). After 10 rounds everyone's grid is filled, forming two 5-digit numbers. A player's score is the sum of her/his two 5-digit numbers. Players with the largest sum win the game. We shall call this the ‘dice-and-numbers game’. It is easy to think of other variants. Any child who is comfortable with column addition and understands how place value affects the size of numbers will develop a reasonable strategy for the game.