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Well done Annie (who didn't give her school) and Rukmini from Dowanhill Primary who sent well explained solutions. Rukmini says:

The totals which were odd were: 1, 3, 3, and 5. There were 4 odd-total dominoes. Erik can only choose these 4 dominoes.
The totals which were even were: 2, 4, 2, 4, and 6. There were 5 even-total dominoes. Sanjay can only choose these 5 dominoes.
In this game, Erik can choose only 4 and Sanjay can choose 5. Sanjay has more, and that's not fair.

In the next game Erik can only choose a multiple of 3, which are: 6, 3 and 3. So Erik can choose 3 dominoes.
And Sanjay can choose 6, so it's not fair again. Sanjay can pick up the most dominoes.

Children from WHS pointed out that it cannot be fair because there are only 9 dominoes and this does not divide equally by two. A very good point, well done.