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Answer: a) 3
a) 12 cups $\rightarrow$ lose one third = 4
$\therefore$ 8 cups left
10 saucers $\rightarrow$ lose half = 5
$\therefore$ 5 saucers left
8 $-$ 5 = 3
$\therefore$ 3 cups do not have matching saucers.
Answer: b) 3, 4, 5, 6 or 7
b) Represent the 12 cups and 10 saucers as being different colours:
If the 4 cups smashed are the bottom 4 and the 5 saucers smashed are the bottom 5 (for example), then there are 3 cups without matching saucers.
If the 4 cups smashed are the top 4 and the 5 saucers smashed are the bottom 5 (for example), then there are 7 cups without matching saucers (see below).
Thank you to Luke from Lode Heath School for proposing and solving part (b) of this problem