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Preheat oven to 350 degrees F. Spray an 8-inch square cake pan with cooking spray.
Whisk together the all-purpose and whole-wheat flours, the baking soda and salt. In a small bowl, stir together the granulated sugar, cinnamon and walnuts. In a large bowl, beat the brown sugar, butter and oil until fluffy. If necessary, use the back of a spoon to press out any lumps in the brown sugar. Beat in the eggs, 1 at a time, beating until fully combined. Beat in the vanilla and yogurt.
Add the flour mixture in 2 batches, stirring until just combined.
Spread half of the batter into the prepared pan. Sprinkle half of the nut mixture over the batter and top with the blueberries, gently pressing them into the batter. Spoon the rest of the batter into the pan, smoothing the top. Sprinkle the remaining nut mixture over the cake, pressing gently. Bake until a wooden toothpick inserted in center comes out clean, about 30 to 35 minutes. Let cool slightly and then unmold and allow to cool completely on a cooling rack. Cut the cake into 2-inch squares.