Preheat oven to 350°F/177°C. Prepare four 7x1-inch round pans.
In a small sauce pot, mix together water, sugar and cream of tartar. Heat up until it becomes light caramel in color. Pour these in two of the round pans. Set aside.
In a small bowl, whisk together egg yolks, whole egg, evaporated milk and condensed milk. Pour this over a sieve to remove the bubble. Divide among two of the pans with the caramel at the bottom.
Bake in the preheated oven on a bain-marie for 30-35 minutes or until set. Set aside to cool. Put in the refrigerator to set for a few hours, better if overnight.
For the cake: grease and line two of the round pans, set aside. In a bowl, whisk together cake mix, vegetable oil, water, egg yolks, ube powder and ube flavoring until smooth. In another bowl, whisk egg whites until if forms stiff peaks, fold whipped eggwhites into uber cake batter. Pour into the prepared pans. Bake in the preheated oven for 20-25 minutes or until cooked. Place also in the refrigerator to cool and set.
To assemble cake: Loosen the flan using a very thin spatula. Place the cake on the pan with flan then tip this over a serving dish. Do the same for the other cake and flan. Place one layer on top of the other to make 4 layers of cake. Serve chilled.