8.3 Flexibility of Closure: Finding a Known Shape Hidden in a Figure
Key Takeaways
IOS defines flexibility of closure as the ability to identify or detect a known pattern, such as a figure, word or object, hidden in other material.
IOS's sample asks which of four objects is included in a figure; only one is, and it may be larger, smaller or rotated.
Size and rotation can change, but proportions cannot, so compare angles and side ratios rather than absolute size.
Search the figure for each candidate's most distinctive feature, such as an unusual angle, a curve meeting a straight line or a specific side ratio.
IOS compares the skill to picking out a particular face in a crowd, which relies on features that do not change with position.
Note
OpenExamPrep provides this independent study guide to help prospective law enforcement candidates prepare for the National Criminal Justice Officer Selection Inventory (NCJOSI). OpenExamPrep is an independent educational publisher and is not affiliated with or endorsed by Industrial/Organizational Solutions (IOS) or any municipal civil service commission.
What IOS Measures
IOS defines flexibility of closure as the ability to identify or detect a known pattern (such as a figure, word or object) that is hidden in other material. The task is to pick out the disguised pattern from the background. IOS adds that the ability involves finding an object hidden within a group of other objects, and compares it to picking out a particular face in a crowd of people or an object in a figure.
The Item Format
IOS's sample items show a line drawing and four answer options, each a small object. The instructions are precise:
- Select the object that is included in the figure.
- Only one of the four objects is included.
- The object may appear larger, smaller or rotated in the figure compared with the answer option, but the proportions will be identical.
That last rule is the key to the whole dimension. You cannot match by size, and you cannot assume the object is upright. You can match by shape: the angles at each corner, the number of sides, which sides are curved, and the ratios between side lengths.
Why Hidden Figures Are Hard
Your visual system groups lines into the most obvious whole shapes. In a drawing made of overlapping polygons, you naturally see the big outlines and not the small shape formed where two of them cross. Flexibility of closure means overriding that automatic grouping and looking for a shape that is "disguised" by lines continuing past its corners.
Two common mistakes follow from this:
- Looking for the object at its printed size. The hidden version may be twice as large or half as large.
- Looking for the object upright. It may be rotated 90 degrees, 45 degrees or upside down.
A Feature-First Search Method
- Describe each candidate in features, not pictures. For each of the four options, note: number of sides, any right angles, any acute or obtuse angles, any curves, and the ratio of the longest side to the shortest.
- Find the rarest feature. A 60-degree corner, a curve joining a straight line, or a long, thin 1:3 rectangle is easier to search for than a plain square.
- Scan the figure for that feature only. Ignore everything else. Look especially where lines cross, because hidden shapes are usually bounded by parts of several longer lines.
- Verify proportions. When you find a candidate location, check that the side ratios match. A 1:2 rectangle cannot match a 1:3 rectangle, however similar they look.
- Eliminate. If a candidate's rarest feature does not exist anywhere in the figure (for example, the figure has no curves at all), that option is out.
Worked Example (Described in Words)
Imagine a figure: a square with both diagonals drawn, and a circle centered inside the square. The four options are:
| Option | Object | Present in the figure? | Reason |
|---|---|---|---|
| a | Equilateral triangle | No | The figure has only 45° and 90° angles; an equilateral triangle needs 60° corners |
| b | Rectangle twice as long as it is wide | No | No line divides the square into a 1:2 rectangle |
| c | Right isosceles triangle | Yes | Each diagonal cuts the square into right isosceles triangles, and the two diagonals form four smaller ones |
| d | Regular hexagon | No | The figure has no 120° corners |
The right isosceles triangle appears in two sizes and four orientations, which is exactly why IOS warns that the hidden object may be larger, smaller or rotated. Its proportions (two equal legs and a right angle) never change.
Faces in a Crowd
IOS's real-world comparison is spotting one face in a crowd. The same principle applies: rely on proportions and features that do not depend on position, such as the spacing of the eyes, the length of the nose relative to the face, and the shape of the jaw. These stable features are covered further in Section 8.4 (Visualization).
Flexibility of Closure vs. Visualization and Selective Attention
| Dimension | The Object Is... | You Must... |
|---|---|---|
| Flexibility of closure | Unchanged in shape, but hidden among other lines and possibly resized or rotated | Find it inside a busy figure |
| Visualization | Changed (moved, rotated, folded or altered in appearance) | Imagine how it looks after the change |
| Selective attention | A string of characters with no changes allowed | Find the one exact copy |
Practice Habits
- Draw two or three overlapping shapes yourself, then search for triangles and quadrilaterals formed by their crossings.
- Use printed hidden-figure puzzles and time yourself; aim to eliminate wrong options rather than to "see" the answer instantly.
- When stuck, trace the candidate's outline with your pencil in the air over the figure, rotating it as you go.
A figure shows a rectangle that is twice as long as it is high, with one diagonal drawn from its lower-left corner to its upper-right corner. No other lines are drawn. Under IOS's rule that the hidden object may be larger, smaller or rotated but keeps identical proportions, which object is included in the figure?
An equilateral triangle
A square
A right isosceles triangle, rotated 45 degrees and enlarged
A right triangle whose two legs are in a 1:2 ratio
One answer option in a flexibility of closure item is a rectangle measuring 1 unit by 3 units. Which rectangle, if found in the figure, could be that same object?
A rectangle measuring 2 units by 6 units, turned on its side
A rectangle measuring 2 units by 4 units, turned upside down
A square measuring 3 units by 3 units
A rectangle measuring 1 unit by 2 units
Which statement correctly describes NCJOSI² flexibility of closure items according to the IOS study guide?
Several of the four objects usually appear in the figure, and you choose the largest one
The objects must appear in the figure at exactly the size and orientation printed in the options
Exactly one object appears, possibly larger, smaller or rotated, but with identical proportions
The items test memory, because the figure is removed before the options are shown
Sections you finish are checked off in the contents.