## 7th and 8th Grade, October, 2017

C.1145. Similar to a magic square, we made a magic 3x3x3 cube by writing the numbers from 1 to 27 on the unit cubes so that the sum of the numbers in every 1×3 column or row is the same. (Not the diagonals!) If you made vertical 3×3 slices of the cube, you would see the following:

How did we place the rest of the numbers on the cube?
C.1146. There are 52 cards in a deck of French cards. There are 4 suits: spades (black leaf), clubs (black leaf of clover), hearts (red heart), and diamonds (red rhombus). In each suit there are J, Q, K and A cards and the numbered cards go from 2 to 10. We took 5 cards out of them: at least one out of every suit; all 5 cards are numbered cards; the sums of the numbers on the odd- and on the even-numbered cards are the same; the sum of the spade cards is 14; the sum of the red cards is 10; the smallest value card is a heart. Which 5 cards did we take out?
C.1147. Are there 21 different positive whole numbers such that the sum of their reciprocals is 1?
C.1148. How many positive 3-digit whole numbers are there which are less than twice the product of their digits?
C.1149. Frank and Bonnie bought the same package of letter papers and envelopes. Frank used 1 sheet, and Bonnie used 3 sheets of paper for each letter. Frank used up all of the envelopes and had 50 sheets left over, Bonnie used up all the sheets and had 50 envelopes left over. How many sheets and how many envelopes were there in the package?
C.1150. Find the smallest positive whole number whose half is a square number, and its fifth is a cube number.
C.1151. Find the smallest positive prime number that can be written as the sums of 2, 3, 4, and even 5 different prime numbers.
C.1152. The diagram below is a number pyramid. Take the sums of the products of the numbers in each row, starting with the first row. Can we ever get a square number (greater than 1) when we stop at any row? (The product of the numbers in the first row is 1.)