Have Chocolate, Will Travel

Algebra Level 3

A traveler visits each point ( x , y ) (x,y) with integer coordinates where 0 ≤ x , y ≤ 4 0\leq x,y \leq 4 (each blue dot in the picture). At each of these points ( x , y ) (x, y) , the traveler eats 1 2 max ⁡ { x , y } + 1 \frac {1}{2\max\{x,y\} + 1}

chocolate bar(s). How many chocolate bars will the traveler eat in all?

Notation . max ⁡ { x , y } \max\{x,y\} represents the largest value in the set { x , y } \{x,y\} .


The answer is 5.

Damon Demas by Damon Demas

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1 solution

Let's go coordinate wise! ( 0 , 0 ) → 1 (0,0) → 1

( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) → 3 × 1 3 (0,1) , (1,0), (1,1)→ 3\times\frac{1}{3}

( 2 , 0 ) , ( 2 , 1 ) , ( 0 , 2 ) , ( 1 , 2 ) , ( 2 , 2 ) → 5 × 1 5 (2,0), (2,1), (0,2),(1,2), (2,2) → 5\times\frac{1}{5}

( 3 , 0 ) , ( 3 , 1 ) , ( 3 , 2 ) , ( 0 , 3 ) , ( 1 , 3 ) , ( 2 , 3 ) , ( 3 , 3 ) → 7 × 1 7 (3,0),(3,1),(3,2),(0,3),(1,3),(2,3),(3,3)→ 7\times\frac{1}{7}

( 4 , 0 ) , ( 4 , 1 ) , ( 4 , 2 ) , ( 4 , 3 ) , ( 0 , 4 ) , ( 1 , 4 ) , ( 2 , 4 ) , ( 3 , 4 ) , ( 4 , 4 ) → 9 × 1 9 (4,0),(4,1),(4,2),(4,3),(0,4),(1,4),(2,4),(3,4),(4,4)→9\times\frac{1}{9}

→ 1 + 3 3 + 5 5 + 7 7 + 9 9 = 5 \large→ 1 + \frac{3}{3} + \frac{5}{5} + \frac{7}{7} + \frac{9}{9} = 5

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