For the **Sum–Sum** game, the goal is to make the left side as big as possible but still smaller than the right side. Since we must use the numbers 1 to 8 once each, we need to balance both sides carefully. One good arrangement is **42 + 53 < 76 + 81**. When we add these, the left side equals 95 and the right side equals 157, which makes the inequality true because 95 is less than 157. The total score is 252, which is the highest possible score for this version of the game.
For the **Take–Take** game, we are subtracting on both sides, so we want the left difference to be smaller and the right difference to be larger. To make a small difference, we can use numbers that are close together, and to make a large difference, we can use one large number and one small number. For example, **65 – 64 < 87 – 21** works perfectly. The left side equals 1, and the right side equals 66, so the inequality is true. This gives a total score of 67, which is the highest possible score for the Take–Take version.
In the **Take–Sum** game, we are subtracting on the left and adding on the right. To make the inequality true, we should make the left side small and the right side large. A good example is **65 – 64 < 87 + 32**. On the left side, the answer is 1, and on the right side, it’s 119. Since 1 is smaller than 119, the inequality works. The total score for this version is 120, which is the highest possible score.
For the **Sum–Take** game, we are adding on the left and subtracting on the right. The problem here is that the largest difference we can make by subtracting (for example, 87 – 12 = 75) is still smaller than the biggest total we can make by adding (such as 76 + 54 = 130). That means it’s impossible to make the left side smaller than the right side when using the numbers 1 to 8. So, the **Sum–Take** version cannot be solved.