Anna's mother has organised a Christmas Party for Anna and nine of her friends.
They are seated at a circular table set with ten places. Each of the ten places has a numbered card, the numbers being the ten integers from zero to nine. Anna's place at the table is numbered zero and the other nine cards are in the other nine places, but not in numeric order.
The age of each of the ten children seated at the table is the sum of the cards on their immediate left- and right-hand sides.
Anna's age and the child sitting opposite her total 25.
The ages of the child to Anna's left and the child sitting opposite her total 13.
The ages of the child two places to Anna's left and the child sitting opposite her total 19.
The ages of the child three places to Anna's left and the child sitting opposite her total 23.
The child three places to Anna's left is four years older than Anna.
Poppy, one of the twins, is sitting on Anna's immediate right.
Holly, the other twin, is sitting opposite Poppy.
The other guests, in order of age with oldest first, are Billy, Ed, John, Becky, Mike, Iwan and Rob.
Starting with Anna and going clockwise around the table, how are the children seated?
The sum of the ages of the ten children is 2 x (1 + 2 + . . . + 9) which is 90.
The sum of the ages of the four pairs of children given is 25 + 13 + 19 + 23, which is 80.
Thus the total ages of the other pair, who are twins, is 10, and so each of the twins is 5.
Poppy (a twin) is sitting on Anna's right, and Anna's card is 0, so the card to the right of Poppy must be 5.
Designate the cards anticlockwise around the table, starting with Anna as:
0 A 5 B C D E F G H
and let the ages of Anna and the three girls to her left be x, y, z and x + 4.
From this we can get the following equations:
A + H = x, G = y, H + F = z, E + G = x + 4, D + F = 5, E + C = 25 - x, D + B = 13 - y, 5 + C = 19 - z and A + B = 19 - x
From these:
B + D + G = 13, H + F + C = 14 and A + B + E + G = 23
So
A + 2B + C + D + E + F + 2G + H = 50 and
A + B + C + D + E + F + G + H = 45 - 5 = 40
So B + G = 10 and D = 3, and then F = 2, H + C = 12 and A + E = 13
Now the cards 0, 2, 3 and 5 are accounted for and 1, 4, 6, 7, 8 and 9 still remain.
H + C = 12 and must be 4 and 8,
B + G = 10 and must be 1 and 9 and
A + E = 13 and must be 6 and 7
From this C = 8, G = 9 and E = 6, and the full list of cards is A = 7, B = 1, C = 8, D = 3, E = 6, F = 2, G = 9 and H = 4.
So around the table from Anna going clockwise we have Becky, Iwan, Billy, Holly, Ed, Rob, John, Mike and Poppy.