1994 AJHSME Problems/Problem 15

Problem

If this path is to continue in the same pattern:

[asy] unitsize(24); draw((0,0)--(1,0)--(1,1)--(2,1)--(2,0)--(3,0)--(3,1)--(4,1)--(4,0)--(5,0)--(5,1)--(6,1)); draw((2/3,1/5)--(1,0)--(2/3,-1/5)); draw((4/5,2/3)--(1,1)--(6/5,2/3)); draw((5/3,6/5)--(2,1)--(5/3,4/5)); draw((9/5,1/3)--(2,0)--(11/5,1/3)); draw((8/3,1/5)--(3,0)--(8/3,-1/5)); draw((14/5,2/3)--(3,1)--(16/5,2/3)); draw((11/3,6/5)--(4,1)--(11/3,4/5)); draw((19/5,1/3)--(4,0)--(21/5,1/3)); draw((14/3,1/5)--(5,0)--(14/3,-1/5)); draw((24/5,2/3)--(5,1)--(26/5,2/3)); draw((17/3,6/5)--(6,1)--(17/3,4/5)); dot((0,0)); dot((1,0)); dot((1,1)); dot((2,1)); dot((2,0)); dot((3,0)); dot((3,1)); dot((4,1)); dot((4,0)); dot((5,0)); dot((5,1)); label("$0$",(0,0),S); label("$1$",(1,0),S); label("$2$",(1,1),N); label("$3$",(2,1),N); label("$4$",(2,0),S); label("$5$",(3,0),S); label("$6$",(3,1),N); label("$7$",(4,1),N); label("$8$",(4,0),S); label("$9$",(5,0),S); label("$10$",(5,1),N); label("$\vdots$",(5.85,0.5),E); label("$\cdots$",(6.5,0.15),S); [/asy]

then which sequence of arrows goes from point $425$ to point $427$?

[asy] unitsize(24); dot((0,0)); dot((0,1)); dot((1,1)); draw((0,0)--(0,1)--(1,1)); draw((-1/5,2/3)--(0,1)--(1/5,2/3)); draw((2/3,6/5)--(1,1)--(2/3,4/5)); label("(A)",(-1/3,1/3),W); dot((4,0)); dot((5,0)); dot((5,1)); draw((4,0)--(5,0)--(5,1)); draw((14/3,1/5)--(5,0)--(14/3,-1/5)); draw((24/5,2/3)--(5,1)--(26/5,2/3)); label("(B)",(11/3,1/3),W); dot((8,1)); dot((8,0)); dot((9,0)); draw((8,1)--(8,0)--(9,0)); draw((39/5,1/3)--(8,0)--(41/5,1/3)); draw((26/3,1/5)--(9,0)--(26/3,-1/5)); label("(C)",(23/3,1/3),W); dot((12,1)); dot((13,1)); dot((13,0)); draw((12,1)--(13,1)--(13,0)); draw((38/3,6/5)--(13,1)--(38/3,4/5)); draw((64/5,1/3)--(13,0)--(66/5,1/3)); label("(D)",(35/3,1/3),W); dot((17,1)); dot((17,0)); dot((16,0)); draw((17,1)--(17,0)--(16,0)); draw((84/5,1/3)--(17,0)--(86/5,1/3)); draw((49/3,1/5)--(16,0)--(49/3,-1/5)); label("(E)",(47/3,1/3),W); [/asy]

Solution

Notice the pattern from $0$ to $4$ repeats for every four arrows. Any number that has a remainder of $0$ when divided by $4$ corresponds to $0$.

The remainder when $425$ is divided by $4$ is $1$. The arrows from point $425$ to point $427$ correspond to points $1$ and $3$, which have the same pattern as $\boxed{\text{(A)}}$.

See Also

1994 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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