1. The problem involves a pattern of hexagonal numbers with each top-right number related to the numbers in the left columns and the middle numbers. 2. To find the rule, observe each diagram: - First: Left = 3,4,9,5,6; middle = 13; top-right = 99 - Second: Left = 6,8,2,7; middle = 20; top-right = 47 - Third: Left = 8,8,2,7,8; middle = 43; top-right = 88 - Fourth: Left = 8,7,5,6; middle = 5; top-right = 50 3. A possible hypothesis is the top-right number is the sum of the products of the pairs formed by left numbers and middle number, or some combination. 4. Check if top-right equals sum of left column numbers plus middle times some factor: - Sum first left set: $3+4+9+5+6=27$ - Sum second left set: $6+8+2+7=23$ - Sum third left set: $8+8+2+7+8=33$ - Sum fourth left set: $8+7+5+6=26$ 5. Check relations: - For first: $13 \times 6 + 27 = 78 + 27 = 105$ (not 99) - For second: $20 \times 1 + 23 = 43$ (not 47) - For third: $43 + 33 = 76$ (not 88) - For fourth: $5 \times 4 + 26 = 46$ (not 50) 6. Try sums of products: - Find products of pairs in left multiplied by middle number: For first: could relate to $13 \times (3+9+6) = 13 \times 18 = 234$ - too large. 7. Without more explicit rules, the pattern is complex and involves summing or multiplying specific combinations. 8. Since exact relation is not determined, one suggestion is that the top-right number is some rule based on sums and multiplications of left numbers and the middle number. Final answer: The numbers follow a pattern involving sums and multiples of the left column and middle numbers, but the exact formula cannot be uniquely determined from the given data.