Chapter 29
Dot Situation and Figure Formation
Give every dot a signature, and give every set of pieces three checks. The answer appears on its own.
Dot situation and figure formation are two small non-verbal topics, and both reward a calm, step-by-step method. They are not asked in every paper, but when they come they are among the quickest marks of the paper.
In a dot situation question, figure X shows a few shapes (circles, triangles, squares, an outer rectangle) with a dot, several dots or some letters placed in them. The four options show the same shapes with the marker placed somewhere else. You have to find the option in which the marker lies in the same kind of region as in X. The whole topic runs on one idea: give every dot a signature, one letter for each shape, Y for inside and N for outside. Two dots with the same signature are in the same kind of region.
In figure formation the pieces (triangles, rectangles, trapezoids) are given in a box, and you choose the figure that can be built by joining all of them, or the numbered pieces that build a given figure. Here we use three checks in a fixed order: area, then angles, then fit. Most options die at the first two checks, before you have to move a single piece in your head.
We first build the basics: the signature, what 'inside' means, the outer frame and the conditions for joining pieces. Then five types, in the order to learn them:
- Type 1, Single Dot: the core form. The dot is in the common region, with or without an exclusion.
- Type 2, Outer Frame: the frame is one more shape; dots in the empty space or outside it.
- Type 3, Letters, Symbols and Several Dots: every marker must be right at the same time.
- Type 4, Figure Formation: pieces to figure, by area, angles and fit.
- Type 5, Piece Selection: figure to pieces, with area first and corners next.
Each type gives the logic first, then the method, then the Kavach Tricks, and a solved example with a figure.
The Signature: Y or N for Every Shape
A dot situation question looks like a picture puzzle, but it is really a small table. Look at the shapes one at a time and ask one question for each: is the dot inside this shape? Write Y for yes and N for no. The letters you collect, in a fixed order of the shapes, are the dot's signature.
[[fig:0]]
In the figure there are two shapes, a circle and a square, and the signature is written in the order circle, square. The dot in the circle only is YN. The dot in the part shared by both is YY. The dot in the square only is NY, and a dot outside both is NN. So two overlapping shapes make 4 regions, and three overlapping shapes make up to 8 regions (7 inside the shapes and one outside all of them).
Now the whole method is one sentence: find the signature of the dot in X, then choose the option whose dot has exactly the same signature. The shapes are drawn identically in X and in the options, so the letters mean the same thing everywhere. If a dot has the same signature in all four options, it cannot help you; skip it and look at another dot.
| Shapes in the figure | Possible regions (including outside all) |
|---|---|
| 2 overlapping shapes | 4: only first, only second, both, neither |
| 3 overlapping shapes | up to 8: three single, three pairs, all three, none |
Inside Means Inside: Lines, Frames and Letters
Before we go to the types, four small conventions that save marks.
1. Inside means strictly inside. A dot sitting on a line belongs to neither side. Well-made questions do not put a dot on a line, so a dot that clearly lies on one side is judged by that side. If you ever meet a dot that touches a boundary, read the question: treat it as not inside unless the question says otherwise.
2. The outer frame is a shape. In dot situation, an outer rectangle (a frame) counts as one more shape. A dot can be inside the frame and outside every inner shape, or outside the frame altogether. In the figure below the signature order is frame, circle: the dot in the circle is YY, the dot in the frame but outside the circle is YN, and the dot beyond the frame is NN.
[[fig:0]]
3. Letters are dots. When letters such as P, K and T are placed in the shapes, each letter has its own signature and is handled exactly like a dot. The only extra point: every letter must keep its own region, so P cannot take K's place.
4. Several dots. Every dot has its own signature, and an option is correct only when all of them are matched together.
A dot in an overlap is inside both shapes; a dot in a part that belongs to one shape only is outside the others.
Three Shapes, Eight Regions
When three shapes cross each other, the picture looks crowded, yet it is only eight signatures. Let the shapes be taken in the order blue circle, green circle, red circle. Then:
- YNN, NYN, NNY: inside one circle only.
- YYN, YNY, NYY: inside exactly two circles (the three petals of the overlap).
- YYY: inside all three (the small middle part).
- NNN: outside all three.
[[fig:0]]
Notice what the figure teaches. The middle region exists only when all three shapes really overlap; in many questions the shapes are of different kinds (a circle, a triangle, a square) and the common region may be small or oddly shaped. So never assume it. Always check by Y and N.
A useful habit: when the dot in X sits in a crowded part, count how many Y letters its signature has. A dot with three Y letters can only lie in the very middle; a dot with one Y lies in a 'tip'. The number of Y letters often tells you at a glance which options to look at first.
Figure Formation: Area, Angle, Fit
In figure formation the pieces are shown in a box, and the answer is the figure made by joining all of them. The exam usually does not print the conditions, so know them: a piece may be turned, but not flipped over, cut, bent or overlapped, and every piece must be used whole, exactly once. If a question seems to need a flip or a cut, read it again; a right-angled triangle that is cut by a line is no longer the same piece.
[[fig:0]]
Two necessary checks come before any fitting:
- Area. The areas of the pieces must add up to the area of the figure. If they do not, the option is wrong at once.
- Angles. Every corner of the figure must be made from the pieces' corners. If the pieces have only 45° and 90° angles, they can make only 45°, 90°, 135°, 180° and so on. A figure with a 39° or a 27° corner cannot be built.
Then comes the fit: place the most unusual piece first (the one with a slanting edge) and see which options can hold it. Area and angles are necessary, but only the fit is sufficient. Two options may pass both checks and differ only at a join, so check the touching edges.
One caution: pieces may be hidden inside a joined figure, so counting the pieces you can see in an option is only a rough filter, not proof.
Type 1
Single Dot: Common Region and Exclusion
Almost every dot situation question starts here. Figure X shows two or three overlapping shapes and one dot. The four options draw the same shapes in the same places, with the dot moved. You must choose the option whose dot lies in the same kind of region as the dot in X.
Why does this work? The shapes never move, so the only thing that can change is the dot. Give the dot a signature: one letter for each shape, Y if the dot is inside it and N if it is outside. Two dots lie in the same kind of region exactly when their signatures are identical.
There are two forms. In the first form the signature is all Y (YY or YYY). Think of the shapes as sheets of coloured glass laid one over another: the dot sits where you look through every sheet at once. In the second form the signature is mixed, such as YYN: the dot is in the circle and the triangle but outside the square. The valid region is the common part of the Y shapes minus the N shapes.
[[fig:0]]
In the figure (order: triangle, square, circle) the dot marked YYY is in all three. The dots marked NYY and YYN are the nearly-right dots that the paper-setter uses as wrong options.
[[fig:1]]
In this second figure (order: circle, triangle, square) the dot YYN is inside the circle and the triangle but outside the square. The dot YYY is in the same overlap, yet it also lies in the square, so it fails the single N.
Step method. Step 1: write the signature of the dot in X. Step 2: find the common part of the Y shapes on X. Step 3: for each N shape, remember that the dot must be outside it. Step 4: go through the options and reject each one at its first wrong letter.
Y means the dot must be inside; N means the dot must be outside. Both are conditions, and two Y out of three is not enough.
Type 2
Outer Frame and Outside All Shapes
In many dot situation figures the shapes sit inside an outer rectangle, a frame, and the examiner asks about a dot that is in the empty part of the frame or even outside it. The logic is the same signature idea; the only new point is that the frame needs a letter of its own.
Many dot situation figures are drawn inside an outer rectangle, a frame. Here is the point that surprises students: the frame is a shape too. In a classification question the frame would be only a border, but in dot situation it takes a letter in the signature. So a dot can be:
- inside the frame and inside an inner shape (for example YY),
- inside the frame but outside every inner shape (YNN), or
- outside the frame altogether (NNN), which also makes it outside every inner shape.
[[fig:0]]
In the figure the signature order is frame, triangle, square. The dot marked YYY is in the overlap of the triangle and the square, and in the frame. The dot marked YYN is in the triangle only (inside the frame). The dot marked YNN is in the frame only: it is outside both inner shapes. The dot marked NNN is outside the frame, so it is outside everything.
The common trap lies in the words 'outside all the shapes'. A dot that is inside the frame but outside the circle and the square is not the same as a dot outside the frame. If the dot of X is outside the frame, the right option has its dot outside the frame as well; do not pick a dot that is merely 'in the empty space' of the frame.
Treat the frame as one more shape in the signature, and the question becomes the same as every other dot question. When the frame is not drawn, there is no extra letter, so read the figure before you decide.
Type 3
Letters, Symbols and Several Dots
Here the markers are letters or symbols (P, K, T, a star, a cross), or there are two or three plain dots in X. The question is the same as before, but now one slip decides the answer: an option may match the first marker and fail the second.
The logic is simple. Every letter and every dot has its own signature, and the correct option must hold all of them at the same time. For letters, each letter must stay in its own kind of region: P cannot sit in K's region even if that region is a perfectly good one. For plain dots, which carry no labels, we ask for the same set of signatures, in any order.
[[fig:0]]
In the figure (order: circle, square) the signatures are P = YY, K = YN and T = NY. The right option has P in the overlap, K in the circle-only part and T in the square-only part.
Why do examiners use letters? The wrong options are easy to make: they simply exchange two letters. Such an option occupies exactly the same regions as X and is still wrong, so the check must be done letter by letter, not region by region.
Step method. Step 1: write the signature of every marker of X. Step 2: take the marker with the rarest signature first (all Y, or all N). Few places can hold it, so it removes the most options. Step 3: check the next marker only in the survivors. Step 4: count: the option must have a partner for each marker, with none left over.
[[fig:1]]
The second figure (order: circle, triangle, square) has two unlabelled dots, YYN and NYN. The option must have one dot of each kind.
Never stop after the first marker. An option that satisfies two markers and misses the third is still wrong.
Type 4
Figure Formation (Piece Assembly)
Here a box holds a few separate pieces: triangles, rectangles, trapezoids. Four figures are given, and you choose the one that can be built by joining all the pieces, each once and whole. The pieces may be turned but not flipped, cut or overlapped.
Do not start by moving pieces in your head. The method is a funnel of three checks, and most options fall through it before you try a fit.
Check 1: area. Add the areas of the pieces. Work out the area of each option figure. An option whose area is different is out. This one check alone often leaves a single option.
Check 2: angles. Every corner of the option must be made from the corners of the pieces. A piece with a 45° corner and a 90° corner can build 45°, 90°, 135°, 180° and so on, but never 27°, 39° or 63°. An option that has the right area but a corner of a different size cannot be built. This is the trap the examiner sets with the second-best option.
Check 3: fit. For the one or two options that survive, place the most distinctive piece first (a slanting piece or the largest one) and see how the rest must lie beside it.
[[fig:0]]
In the figure a rectangle 3 by 2 and two small right-angled triangles are placed so that the triangle's right angle sits on the rectangle's corner; the bottom edge becomes 2 + 3 + 2 = 7 and the top stays 3.
Area first, angles second, fit third. If the pieces include curved parts (quarter circles), the same idea works with matching edges: curved edge meets curved edge.
Type 5
Piece Selection
This is figure formation run backwards. A target figure is given, together with several numbered pieces, and the options are sets of numbers such as "1, 3 and 5". You choose the set whose pieces make the target exactly.
The method is the same funnel, with one extra tool, the target's own corners and edges.
Step 1: area of the target. Work it out once.
Step 2: area of each piece. Write the number next to each piece.
Step 3: add the areas of each option set. A set with a different total is out. Many options die here.
Step 4: corners and edges. This is the check that finishes the question, because two sets often have the same total. Look at the corners of the target: a 45° corner can be filled only by a piece that has a 45° corner, and a slanting side needs a slanting piece. A set made only of rectangles and squares has no slanting edge and cannot make a figure with a slanted side.
[[fig:0]]
In the figure two triangular pieces (1 and 2) join along their long sides to make a square of side 4. Notice what the check says. Pieces such as a square or a rectangle have the right sum only by chance; they cannot fill a corner of 45°.
Area is necessary but not sufficient. A set with the right total still fails if it has no piece that can fit the target's slanting edge or sharp corner. In the exam, use area to cut the options to two, and the corners to cut the last one.