Ten children, $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots , c _ { 9 }$, are seated clockwise in a circle. The teacher walks clockwise behind the children with a large bag of sweets. She gives a sweet to child $c _ { 1 }$. She then skips a child and gives a sweet to the next child, $c _ { 3 }$. Next she skips two children and gives a sweet to the next child, $c _ { 6 }$. She continues in this way, at each stage skipping one more child than at the preceding stage before giving a sweet to the next child. (i) The $k$ th sweet is given to child $c _ { i }$. Explain why $i$ is the last digit of the number $$\frac { k ( k + 1 ) } { 2 } .$$ (ii) Let $1 \leqslant k \leqslant 18$. Explain why the $k$ th and ( $20 - k - 1$ )th sweets are given to the same child. (iii) Explain why the $k$ th sweet is given to the same child as the $( k + 20 )$ th sweet. (iv) Which children can never receive any sweets? When the teacher has given out all the sweets, she has walked exactly 183 times round the circle, and given the last sweet to $c _ { 0 }$. (v) How many sweets were there initially? (vi) Which children received the most sweets and how many did they receive?
(i) [2 marks] Each time the teacher gives out a sweet, the number of children skipped increases by 1 , so the $k$ th sweet will be given after $\frac { k ( k + 1 ) } { 2 }$ steps. However, as there are 10 children in the circle, we only need to keep track of the unit digit, since every 10 steps the unit digit will return to the original unit digit. [0pt]
\section*{5. For ALL APPLICANTS.}
Ten children, $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots , c _ { 9 }$, are seated clockwise in a circle. The teacher walks clockwise behind the children with a large bag of sweets. She gives a sweet to child $c _ { 1 }$. She then skips a child and gives a sweet to the next child, $c _ { 3 }$. Next she skips two children and gives a sweet to the next child, $c _ { 6 }$. She continues in this way, at each stage skipping one more child than at the preceding stage before giving a sweet to the next child.\\
(i) The $k$ th sweet is given to child $c _ { i }$. Explain why $i$ is the last digit of the number
$$\frac { k ( k + 1 ) } { 2 } .$$
(ii) Let $1 \leqslant k \leqslant 18$. Explain why the $k$ th and ( $20 - k - 1$ )th sweets are given to the same child.\\
(iii) Explain why the $k$ th sweet is given to the same child as the $( k + 20 )$ th sweet.\\
(iv) Which children can never receive any sweets?
When the teacher has given out all the sweets, she has walked exactly 183 times round the circle, and given the last sweet to $c _ { 0 }$.\\
(v) How many sweets were there initially?\\
(vi) Which children received the most sweets and how many did they receive?