### Brain Teasers

# Lotto Luck

Professor Abacus is purchasing a ticket for the Deca Lotto. The lotto number has ten digits, using the numerals 0 through 9, each numeral used once. The clerk asked what number he wanted to pick. Professor Abacus handed the clerk a piece of paper with nine statements, saying "If you can correctly figure out the number, I will give you half of whatever I win." What is the number?

1) The sum of the first five digits is a prime number.

2) The sum of the last five digits is a triangle number*.

3) The sum of the digits in the odd positions is an odd number.

4) The sum of the middle two digits is a square number.

5) The sum of the middle four digits is a cube number.

6) The difference between the 1st and 10th digits is two.

7) The difference between the 2nd and 9th digits is three.

8) The difference between the 3rd and 8th digits is four.

9) The numeral 4 is somewhere in the first five positions.

* You can form a triangle arrangement by building it in the pattern row 1 = 1, row 2 = 2, row 3 = 3 etc. eg. 10 is a classic triangle number as per ten pin bowling. They are arranged in a triangle 1, 2, 3, 4.

1) The sum of the first five digits is a prime number.

2) The sum of the last five digits is a triangle number*.

3) The sum of the digits in the odd positions is an odd number.

4) The sum of the middle two digits is a square number.

5) The sum of the middle four digits is a cube number.

6) The difference between the 1st and 10th digits is two.

7) The difference between the 2nd and 9th digits is three.

8) The difference between the 3rd and 8th digits is four.

9) The numeral 4 is somewhere in the first five positions.

* You can form a triangle arrangement by building it in the pattern row 1 = 1, row 2 = 2, row 3 = 3 etc. eg. 10 is a classic triangle number as per ten pin bowling. They are arranged in a triangle 1, 2, 3, 4.

### Answer

73421058697 + 3 + 4 + 2 + 1 = 17 (prime), 0 + 5 + 8 + 6 + 9 = 28 (triangle), 7 + 4 + 1 + 5 + 6 = 23 (odd), 1 + 0 = 1 (square), 2 + 1 + 0 + 5 = 8 (cube), the difference between 7 and 9 is 2, the difference between 3 and 6 is 3, the difference between 4 and 8 is 4, 4 is in the 3rd position.

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