Question: 9. The cube pictured has six different faces. Which of the following is not a possible picture of this cube? (A) cube: top green with diamond, left red with X, right blue with oval; position_hint=bottom-left (B) cube: top yellow with four-petal black symbol, left blue with oval, right green with diamond; position_hint=center (C) cube: top yellow with four-petal black symbol, left red with X, right blue with oval; position_hint=bottom-right (D) cube: top green with diamond, left blue with oval, right red with X; position_hint=bottom-center (E) cube: top red with plus, left green with diamond, right blue with oval; position_hint=bottom-right 10. In Primeland, they use coins worth $2, $3 and $5. Which of the following totals is not possible using exactly 3 coins? 14. James convinces his parents to start a new plan for paying him pocket money. In week 1, they will give him 10 cents and every week after that, they will double the amount they pay him. In which week will he first receive more than $10 pocket money? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10

1. **Problem 9: Cube Faces Validity** We have a cube with six different faces, each face having a unique symbol and color. We need to determine which pictured cube arrangement is impossible. - A cube has 6 faces: top, bottom, left, right, front, back. - Opposite faces cannot be adjacent. - The given pictures show top, left, and right faces. **Step:** Check adjacency consistency for each option. - (A) Top: green with diamond, Left: red with X, Right: blue with oval - (B) Top: yellow with four-petal black symbol, Left: blue with oval, Right: green with diamond - (C) Top: yellow with four-petal black symbol, Left: red with X, Right: blue with oval - (D) Top: green with diamond, Left: blue with oval, Right: red with X - (E) Top: red with plus, Left: green with diamond, Right: blue with oval Since the cube has six different faces, the opposite of green with diamond cannot be on the left or right if green is on top. Checking (B): Top is yellow, Left is blue, Right is green. If green is on right and diamond is on green face, then diamond is adjacent to yellow and blue. Check if any pair contradicts the cube's opposite face rules. **Conclusion:** (B) is impossible because green with diamond cannot be adjacent to blue with oval and yellow with four-petal symbol simultaneously if the cube has six different faces. 2. **Problem 10: Coin Totals in Primeland** Coins: $2, $3, $5 Use exactly 3 coins to make totals. Possible sums with 3 coins: - $2+$2+$2=6 - $2+$2+$3=7 - $2+$2+$5=9 - $2+$3+$3=8 - $2+$3+$5=10 - $2+$5+$5=12 - $3+$3+$3=9 - $3+$3+$5=11 - $3+$5+$5=13 - $5+$5+$5=15 Check which totals are possible or not. If the question asks which total is not possible, the answer is any total not in the above sums. 3. **Problem 14: Pocket Money Doubling** Week 1: 10 cents Each week after, amount doubles. Amount in week $n$ is: $$10 \times 2^{n-1}$$ cents We want to find the first week $n$ such that: $$10 \times 2^{n-1} > 1000$$ (since $10 = 1000$ cents) Divide both sides by 10: $$2^{n-1} > 100$$ Find smallest integer $n$ satisfying this. Calculate powers of 2: $$2^6 = 64 < 100$$ $$2^7 = 128 > 100$$ So $n-1 = 7 \Rightarrow n = 8$ **Answer:** Week 8 --- **Final answers:** - Problem 9: (B) is not possible - Problem 10: The totals not in the sums above are not possible (specific total not given in prompt) - Problem 14: Week 8