Visual thinking is the term used commonly in psychology to refer to the type of thinking that results from perceiving or processing visual stimuli, forms, or patterns. Such thinking is a major function of the right hemisphere of the brain, also known (logically enough) as the "visual hemisphere."

There are many puzzle genres that entail the use of visual thinking. One involves geometrical figures (triangles, rectangles, etc.) that embed within them a number of similar smaller figures (triangles, rectangles, etc.). The challenge is to identify how many such figures there are in total. The solution strategy calls for visualizing how parts can be assembled to form wholes (a typical right-hemispheric function). Such puzzles are, thus, exercises in "whole-part" thinking. Some people find them to be among the most frustrating of all puzzle types, since it us not unusual to come up with different answers each time one counts the smaller figures one "sees" in the original figure.

I am not sure who invented this puzzle genre, nor from what era of puzzle history it emerges. It is clear, however, that the ancient geometers constructed figures and geometrical patterns with smaller figures (triangles, circles, rectangles, ellipses, etc.) embedded in them in order to study general geometrical principles, to derive theorems, to prove propositions, and so on. They also invented puzzles and games that involved whole-part thinking. One of these was Archimedes' loculus-a 14-piece puzzle forming a square from which it was possible to form different figures (animals, plants etc.) by rearranging the pieces. An online version of the game, with instructions in Latin, can be found on the Bibliotheca Augustana website. Visual thinking puzzles (of all types) seem to be perfect models for testing the validity of what has been called "field dependence theory" in psychology-a notion that originated with the Gestalt psychologists in the 1930s and from which we probably got our expression "not seeing the forest for the trees."

Below are two classic examples of the genre. These take time to solve, so patience is in order. They appear commonly in puzzle collections of all kinds. I should mention that the idea for this blog was kindled by a reader of my book, The Total Brain Workout, who wanted to know how these puzzles were solved. Of course, if you come up with a different solution strategy, or different answers, I would really appreciate knowing about it. As a hint, number the segments you see before starting.

1. How many four-sided sided 900 figures (squares and rectangles) do you see in the diagram below?

2. How many triangles do you see in the diagram below?

Scroll down for the answers!

Answers

The best approach to solving such puzzles, in my view at least, is to number the segments (or embeds) in the figure, and then use the numbers to identify: (1) the "stand-alone" segments (rectangles in this case), and (2) the segments that can be assembled to produce the required figure. The first puzzle can be numerated as follows, and below are the 23 stand-alone segments and assemblages that produce the solution (= 23 rectangles). By the way, the order in which I have laid them out is irrelevant.

Stand-alone rectangles

(1) 1

(2) 2

(3) 3

(4) 4

(5) 5

(6) 6

(7) 7

(8) 8

(9) 9

Assembled rectangles

(10) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 (the overall rectangle)

(11) 1 + 2

(12) 1 + 2 + 4 + 5

(13) 1 + 4

(14) 2 + 5

(15) 3 + 6

(16) 3 + 6 + 7

(17) 3 + 6 + 7 + 8 + 9

(18) 6 + 7

(19) 6 + 7 + 8 + 9

(20) 8 + 9

(21) 4 + 5

(22) 2 + 5 + 3 + 6 + 7 + 8 + 9

(23) 7 + 8 + 9

The solution to the second puzzle is 18 triangles.

Stand-alone triangles

(1) 1

(2) 2

(3) 4

(4) 5

(5) 6

(6) 7

(7) 8

(8) 9

Assembled triangles

(9) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 (large triangle)

(10) 1 + 2 + 3 + 4

(11) 2 + 5

(12) 2 + 5 + 6

(13) 3 + 7

(14) 4 + 8

(15) 4 + 8 + 9

(16) 5 + 7 + 8

(17) 5 + 7

(18) 7 + 8