# Mince Pie Madness Puzzle - Solution

### The Puzzle:

One afternoon, Hungry Horace made a large number of mince pies for his Christmas party the following day. He left them to cool overnight.

During the night Horace's brother Boris came downstairs and divided the mince pies into four equal piles with one left over, which he ate. He also ate one of the four piles, then went back to bed.

Later on, Horace's sister Doris came downstairs and again divided the mince pies into four equal piles with one left over, which she ate. She also ate one of her four piles and returned to bed.

In the morning, Horace found 60 pies left. How many had he made?

During the night Horace's brother Boris came downstairs and divided the mince pies into four equal piles with one left over, which he ate. He also ate one of the four piles, then went back to bed.

Later on, Horace's sister Doris came downstairs and again divided the mince pies into four equal piles with one left over, which she ate. She also ate one of her four piles and returned to bed.

In the morning, Horace found 60 pies left. How many had he made?

### Our Solution:

HORACE MADE 109 MINCE PIES

During the night Boris made four piles of 27 and ate the extra one. He then ate one of the piles. (Total: 28 eaten)

Later Doris divided the remaining 81 pies into four piles of 20 and ate the extra one. She then ate one of the piles.

(Total: 21 eaten)

So in the morning Horace found 60 left!

The equation is:

(3/4) × ( (3/4) × (p-1) - 1 ) = 60

Which can be solved like this:

(3/4) × (p-1) - 1 = 60 × (4/3)

(3/4) × (p-1) = 60 × (4/3) + 1

p-1 = ( 60 × (4/3) + 1 ) × (4/3)

p = ( 60 × (4/3) + 1 ) × (4/3) + 1

p = ( 80 + 1 ) × (4/3) + 1

p = 81 × (4/3) + 1

p = 108 + 1

p = 109

During the night Boris made four piles of 27 and ate the extra one. He then ate one of the piles. (Total: 28 eaten)

Later Doris divided the remaining 81 pies into four piles of 20 and ate the extra one. She then ate one of the piles.

(Total: 21 eaten)

So in the morning Horace found 60 left!

The equation is:

(3/4) × ( (3/4) × (p-1) - 1 ) = 60

Which can be solved like this:

(3/4) × (p-1) - 1 = 60 × (4/3)

(3/4) × (p-1) = 60 × (4/3) + 1

p-1 = ( 60 × (4/3) + 1 ) × (4/3)

p = ( 60 × (4/3) + 1 ) × (4/3) + 1

p = ( 80 + 1 ) × (4/3) + 1

p = 81 × (4/3) + 1

p = 108 + 1

p = 109

Puzzle Author: Stephen Froggatt

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