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1. Anna and Oleg are collecting natural numbers. Anna is collecting only numbers with different digits, and so far she has collected all such numbers up to 1023. Oleg is collecting only prime numbers, and so far he has collected all such numbers up to 2017. What is the largest number which appears in both Anna’s and Oleg’s collection now.

 

2. How many different ways are there to place nine different digits from 1 to 9 inside a three by three grid of nine squares, so that every pair of consecutive digits, their cells share a side. 

 

3. All possible diagonals drawn from the two adjacent vertices A and B of a regular hectogon divide the hectogon’s interior into a number of non-overlapping shapes - triangles and quadrilaterals. How many of these shapes are quadrilaterals?

 

4. I define an RSM - word as a 6 - letter word containing two letters R, two letters S, and two letters M, in which there at least one way that Rosemary can circle three letters such that the circled letters read R-S-M. How many different RSM-words are there? 

supermanaccz  Mar 20, 2018
 #1
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1)

983 should appear on both of their lists, since Anna is collecting numbers with different(non-repeating) digits and Oleg is collecting prime numbers. This is the largest such number up to 1023 that meets both conditions.

Guest Mar 20, 2018
 #2
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4) If I understand your question correctly, here are all the permutations possible. Since the word "R-S-M" can occupy 4 places starting from the left, then you should have 4! = 24 such words. You may count them going across from left to right.

 

{R, R, S, S, M, M} | {R, R, S, M, S, M} | {R, R, S, M, M, S} | {R, R, M, S, S, M} | {R, R, M, S, M, S} | {R, R, M, M, S, S} | {R, S, R, S, M, M} | {R, S, R, M, S, M} | {R, S, R, M, M, S} | {R, S, S, R, M, M} | {R, S, S, M, R, M} | {R, S, S, M, M, R} | {R, S, M, R, S, M} | {R, S, M, R, M, S} | {R, S, M, S, R, M} | {R, S, M, S, M, R} | {R, S, M, M, R, S} | {R, S, M, M, S, R} | {R, M, R, S, S, M} | {R, M, R, S, M, S} | {R, M, R, M, S, S} | {R, M, S, R, S, M} | {R, M, S, R, M, S} | {R, M, S, S, R, M} | {R, M, S, S, M, R} | {R, M, S, M, R, S} | {R, M, S, M, S, R} | {R, M, M, R, S, S} | {R, M, M, S, R, S} | {R, M, M, S, S, R} | {S, R, R, S, M, M} | {S, R, R, M, S, M} | {S, R, R, M, M, S} | {S, R, S, R, M, M} | {S, R, S, M, R, M} | {S, R, S, M, M, R} | {S, R, M, R, S, M} | {S, R, M, R, M, S} | {S, R, M, S, R, M} | {S, R, M, S, M, R} | {S, R, M, M, R, S} | {S, R, M, M, S, R} | {S, S, R, R, M, M} | {S, S, R, M, R, M} | {S, S, R, M, M, R} | {S, S, M, R, R, M} | {S, S, M, R, M, R} | {S, S, M, M, R, R} | {S, M, R, R, S, M} | {S, M, R, R, M, S} | {S, M, R, S, R, M} | {S, M, R, S, M, R} | {S, M, R, M, R, S} | {S, M, R, M, S, R} | {S, M, S, R, R, M} | {S, M, S, R, M, R} | {S, M, S, M, R, R} | {S, M, M, R, R, S} | {S, M, M, R, S, R} | {S, M, M, S, R, R} | {M, R, R, S, S, M} | {M, R, R, S, M, S} | {M, R, R, M, S, S} | {M, R, S, R, S, M} | {M, R, S, R, M, S} | {M, R, S, S, R, M} | {M, R, S, S, M, R} | {M, R, S, M, R, S} | {M, R, S, M, S, R} | {M, R, M, R, S, S} | {M, R, M, S, R, S} | {M, R, M, S, S, R} | {M, S, R, R, S, M} | {M, S, R, R, M, S} | {M, S, R, S, R, M} | {M, S, R, S, M, R} | {M, S, R, M, R, S} | {M, S, R, M, S, R} | {M, S, S, R, R, M} | {M, S, S, R, M, R} | {M, S, S, M, R, R} | {M, S, M, R, R, S} | {M, S, M, R, S, R} | {M, S, M, S, R, R} | {M, M, R, R, S, S} | {M, M, R, S, R, S} | {M, M, R, S, S, R} | {M, M, S, R, R, S} | {M, M, S, R, S, R} | {M, M, S, S, R, R} 

Guest Mar 20, 2018
 #4
avatar+606 
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This may be right if there were no restrictions, but as I stated in the problem, you must be able to circle to letters R-S-M going left to right without anything in between. 

 

The last few example were not correct because R-S-M are not in that order next to one another. 

 

Thanks for trying!

supermanaccz  Mar 21, 2018
 #3
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My dear lord!!! That took you time. 

supermanaccz  Mar 20, 2018

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