Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #20
Grade 6 geometry-3d
Pick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem hands us a 2D top-down map and asks what the 3D structure looks like from the front — the canonical trigger for Tool #17 (Visualize Spatial Relationships). The key spatial insight is that the front view of a column is whichever stack in that column is tallest, because the taller one hides the shorter one. Tool #10 (Create a Physical Representation) is the safety net: if the mental projection is shaky, build the stack from real cubes (or sketch the silhouette column by column on paper) and look at it from the front. Tool #3 (Eliminate Possibilities) is the multiple-choice cross-check: once we know the three column heights, only one of the five pictures has exactly those heights in that order.
Set up the projection rule
Looking from the front collapses each column's two stacks into one silhouette, whose height is the taller of the two.
Reading a top-down grid as (x, y) positions and projecting onto the (x, z) plane is the Grade 5 coordinate-axes idea, extended to a third axis.
When the stacked-cube structure is viewed from the front, the height seen for each column equals the height of the taller of that column's two stacks, because the taller stack hides the shorter one.
▸ Why?
The two stacks in one column share the same left-right position and differ only in depth, so from the front they stand one directly behind the other and their outlines fall on the same part of the picture.
▸ Why?
A front view is formed by looking straight along the back-to-front direction, which records each cube's left-right position and height but not how far back it sits, so two cubes differing only in depth land on the exact same spot.
▸ Why?
Where they overlap, the taller stack blocks the shorter one, and since each stack is cubes piled from the floor upward, the taller pile reaches every level the shorter one does and then higher, so the top edge you see is the taller pile's top.
▸ Why?
A stack of h cubes fills every level from the floor up to height h with no gaps, so a taller stack covers all the levels of a shorter stack behind or in front of it and none of the shorter one shows above it.
Read off the stack map
Pair the back row over the front row column by column, so each column shows its two competing heights.
Laying the two rows on top of each other column by column turns the map into three pairs of heights — exactly the input to the max rule.
5.G.A.1Visualize Spatial RelationshipsTake each column's tallest stack
Take the maximum of each column — the taller stack hides the shorter — giving front-view heights 2, 3, 4.
If you actually stack physical cubes on a table and crouch to look from the front, the only height you can see for each column is the tallest one — the rule writes itself.
6.SP.A.3Create A Physical RepresentationMatch against the choices
Match heights 2, 3, 4 against the five pictures: only (B) is that left-to-right staircase; every other choice misses a column.
Only (B) is the staircase 2-3-4 left to right. (C) has only two columns, and (A), (D), (E) have at least one column whose height does not match — each can be crossed off in one glance.
6.SP.A.3Eliminate PossibilitiesFrom the front, the tallest stack in each column hides the shorter one — take the max of each column and the silhouette 2, 3, 4 is the answer (B).
- Set up the projection rule
- Read off the stack map
- Take each column's tallest stack
- Match against the choices
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