# Tanya’s english cucumber game

Kate went home leaving 3 people to share 2 cucumbers

During the summer holidays Tanya, Tinku, Kate and Peter played together at each other’s place.  They had to play indoors because it was very hot outside.  Wherever they played, they also got something to eat and drink.

Peter’s mom  also gave them gave them two English cucumbers.  She told them to share everything equally amongst themselves.

Kate did not care much for the English cucumber.  So she went home.  That means only 3 kids were there.  Tanya asked Peter to call her mom to bring a knife to cut the two English cucumbers.

Tanya: Aunty, Kate has left and we don’t know exactly how to divide the two English cucumbers equally amongst the three of us.  I have a suggestion.  Please, cut the two English cucumbers each into equal halves first.

Tanya’s game

Aunty did just that.  Each piece was 1/2 an English cucumber (see the picture Tanya’s game with the english cucumber slices).  The three kids took one piece each. So now a piece of half an English cucumber remained. Tanya asked aunty to cut this piece into two equal halves. Now each of the two pieces each was 1/4 English cucumber. She asked aunty to cut each piece the same way again to make a total of 4 pieces each of which was now 1/8 of an English cucumber.  Now that there are 4 pieces – each of the kids took a piece.

Tanya asked Aunty to do the same thing to this remaining piece as she has done with the 1/2 English cucumber piece.  Remember, this was only 1/8 of one English cucumber.  She sliced it to get two pieces each of which is 1/16 of a English cucumber, and then cut- them again.  Now, there were 4 pieces of 1/32 of an English cucumber each.

The three kids took one piece each out of the four with one piece of 1/32 of an English cucumber remaining behind.  Before Tanya could open her mouth, aunty scolded her and said no more cuts.

Tinku:  Did I get my full share which would have been 2/3 of an English cucumber because there were two English cucumbers for the three of us to share.

Tanya: No, because there is a small piece left. You can have it if you want.

Tinku:  I don’t care for the small piece of the English cucumber but I just want to figure everything out.

Tanya:  We got 1/2 an English cucumber each the first time, 1/8 the second time and 1/32 the third time. You can also say that we got a total of 1/2 + 1/8 +1/32 of  an English cucumber each.  This is like saying that we got 16/32 + 4/32 +1/32 English cucumber each.  That means each of us got a total of 21/32 English cucumber. We should have received 2/3 an English cucumber each. A common multiple with 3 and 32 as factors will be 96.  Now 21/32 is equivalent to 63/96 and 2/3 is the same as 64/96.  We got 63/96 of an English cucumber but our share should have been 64/96 each.

Peter thought that he could never get 2/3 of a English cucumber this way.  Tanya just created a good game for his mom who would have given up sooner or later, and the only question was when.  He started giggling and the other two left for home.

Challenge Large Party Pizza

You have a large party pizza with 24 slices.  You and your friends are 10 people to share it equally.  That means each person will get 10/24 of this pizza.  Explain, how this can be done most fairly to everyone.  Your mom is ready to help by cutting any piece into two halves.

Solution: First everybody gets two slices thus 20 slices will give each 2/24 of a pizza.  Remaining are 4 slices – each 1/24 of pizza.  They can be cut into halves to get 8 pieces and then again into halves to get 16 pieces – each piece would be 1/96 of the pizza. Everyone can have a piece and 6 pieces will be left.  Each of the six pieces can be cut into two halves, to get what we will call them tiny pieces 1/192 of the pizza each.  Everyone takes one so that two tiny pieces will be left behind. This way each person will get 2/24 + 1/96 + 1/192 or 16/192 + 2/192 + 1/192 or 19/192 of one pizza. The exact share should be 1/10 or 19.2/192 per person. One can be happy with this or keep cutting again and again.  It is this way because both 2 and 5 are factors of 10 but the cuts are only by the factor 2.

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