Select the alternative which represents three out of the five alternative…

2023

Select the alternative which represents three out of the five alternative figures which when fitted into each other would form a complete square.

  1. A.

    123

  2. B.

    124

  3. C.

    134

  4. D.

    345

Attempted by 2 students.

Show answer & explanation

Correct answer: C

When several figures are said to combine (by rotating them, without resizing or overlapping) into one named shape, three things must hold: every edge lying along the assembled shape's outer boundary must have the same length as it actually has on the target's own outline; at any point strictly inside the shape (not on its boundary) where two or more chosen pieces meet, the angles they contribute there must add up to 360 degrees, while along a straight portion of the boundary the contributed angles add up to 180 degrees, and at one of the target's own corners they must add up to that corner's own angle; and a piece that bulges outward everywhere (convex) can never occupy a spot that the target's own diagram shows folding inward (concave), and vice versa.

  1. The diagram of the assembled square shows two straight cuts, one starting at the top-left corner and one starting at the top-right corner, running inward and ending close together at two distinct points joined by a short connecting edge (not a single shared point). From the right-hand one of those two points, one further straight cut runs all the way down to the bottom-left corner. This divides the square into exactly three four-sided regions: a plain, convex quadrilateral along the top edge (the two top corners plus the two nearby interior points); a concave (inward-folding) quadrilateral along the left edge (the top-left corner, the two interior points, and the bottom-left corner); and a concave quadrilateral wrapping the bottom-right corner (the right-hand interior point, the top-right corner, the bottom-right corner, and the bottom-left corner), which keeps one of the square's genuine 90 degree angles.

  2. Among the five candidate figures, three are convex (figures 1, 2 and 5) and two fold inward at one vertex (figures 3 and 4). Since the square's diagram needs exactly one convex piece and two concave pieces, at most one of figures 1, 2 and 5 can belong to the correct set, and both figures 3 and 4, being concave, are needed to fill the two concave regions.

  3. Comparing the corner angles confirms which convex figure is the right one: figure 1's four angles closely match the top region's angle set, while figure 2's and figure 5's angles do not (figure 2's widest angle is close to a right angle, which the top region does not have; figure 5's genuine right angle sits between two long, nearly equal sides, unlike the top region's four unequal angles).

  4. Figure 3, whose inward (reflex) angle sits next to a true right angle, matches the bottom-right region; figure 4's reflex angle and three smaller angles match the left region.

Rotating figures 1, 3 and 4 into place along the edges they share with the square's boundary reproduces the diagram exactly, with the angles meeting at each internal point adding up to 360 degrees, confirming the fit.

Hence figures 1, 3 and 4 combine to form the complete square.

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