Question

A dice throw can result in the numbers 2,3,5 or 6. Every 4th throw will result in 3. What is the minimum number of times the dice has to be thrown for the pawn to move from zero to reach exactly at 100?

Answer 1

Correct Answer: 5

To solve the problem, we need to determine the minimum number of dice throws required for a pawn to reach exactly 100 on a board with throws resulting in numbers 2, 3, 5, or 6, while every 4th throw always results in a 3.

Let's break down the solution: 

1. Every set of 4 dice throws sums to a fixed value. Calculate this value:

• Assume the first three throws result in the maximum possible: 6.
• The 4th throw is always 3. Therefore, in one complete set of 4 throws, the total score is:
  6 + 6 + 6 + 3 = 21.

1. Calculate how many complete sets of 4 throws are needed to exceed or reach a score of 100:

• The maximum score from each set is 21.
• To find the number of complete sets needed, divide 100 by 21: 
  100 ÷ 21 ≈ 4.76.
• Thus, at least 5 complete sets of throws are initially considered.

1. Calculate the total score after 5 complete sets:

• Total score = 5 sets × 21 points = 105 (exceeds 100).

1. Adjust throws to reach exactly 100:
   • Currently over by 5 (105 - 100 = 5), reduce this in the 20th throw, which must be a 3, so remove 2 more points.
     • Instead of the first maximum 6 in the fifth set, throw a 5.

In conclusion, at least 20 throws are needed to achieve exactly 100:

• 4 sets of 21 (4 × 21 = 84) + 5 (1st throw of 5th set of 5 instead of 6) + 3 + 6 + 6 (for 18 additional points) to hit a total of 100 exactly.