Question

Eva lived in an enchanted village. She loved to go visit her grandmother who lived across the mystic forest. Eva decided to bake a few pies for her grandmother's birthday. But before she could begin her mother told her that the forest path was being watched and guarded by trolls at gates called the "Troll Tolls". They stop travelers and asked them for some form of payment. There were 7 tolls in total and when she got there, she would have to give them half of the cakes that she baked. The trolls were kind enough to usually return one back to the traveler.
How many cakes does Eva have to bake and carry with her in order to have exactly two cakes when she reaches her grandmother's house?​

Answer 1

Solution :-

Let us assume that, total cake eva bake were x .

Now,

→ First troll took = Half of x and return 1 back = (x/2 - 1) cakes .

so,

→ cake left after first troll = x - (x/2 - 1) = (x/2 + 1) again,

→ Second troll took = Half of left and return 1 back = (1/2)(x/2 + 1) - 1 = (x/4 + 1/2) - 1 = (x/4) - (1/2)

→ cake left after second troll = (x/2 + 1) - {(x/4) - (1/2)} = {(x/4) + (3/2)} again,

→ Third troll took = Half of left and return 1 back = (1/2){(x/4) + (3/2)} - 1 = (x/8 + 3/4) - 1 = {(x/8) - (1/4)}

→ cake left after third troll = {(x/4) + (3/2)} - {(x/8) - (1/4)} = {(x/8) + (7/4)}

again,

→ Fourth troll took = Half of left and return 1 back = (1/2){(x/8) + (7/4)} - 1 = {(x/16) + (7/8)} - 1 = {(x/16) - (1/8)}

→ cake left after fourth troll = {(x/8) + (7/4)} - {(x/16) - (1/8)} = {(x/16) + (15/8)}

again,

→ Fifth troll took = Half of left and return 1 back = (1/2){(x/16) + (15/8)} - 1 = {(x/32) + (15/16)} - 1 = {(x/32) - (1/16)}

→ cake left after Fifth troll = {(x/16) + (15/8)} - {(x/32) - (1/16)} = {(x/32) + (31/16)}

again,

→ Sixth troll took = Half of left and return 1 back = (1/2){(x/32) + (31/16)} - 1 = {(x/64) + (31/32)} - 1 = {(x/64) - (1/32)}

→ cake left after Sixth troll = {(x/32) + (31/16)} - {(x/64) - (1/32)} = {(x/64) + (63/32)}

again,

→ Seventh troll took = Half of left and return 1 back = (1/2){(x/64) + (63/32)} - 1 = {(x/128) + (63/64)} - 1 = {(x/128) - (1/64)}→ cake left after Seventh troll = {(x/64) + (63/32)} - {(x/128) - (1/64)} = {(x/128) + (127/64)}

therefore,

→ {(x/128) + (127/64)} = 2

→ (x/128) = 2 - (127/64)

→ (x/128) = (128 - 127)/64

→ (x/128) = (1/64)

→ x = 2 (Ans.)

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