10 Puzzles to Test Your Thinking in Multiple Ways

Knowledge, logic, and number sense will all be needed.

Posted Aug 28, 2016

bikeriderlondon/Shutterstock
Source: bikeriderlondon/Shutterstock

I have shared puzzles based on arithmetic in previous posts. They generated quite a bit of correspondence, acknowledging both their challenging nature and the possibility of different solutions to the ones I provided. This post revisits this area of puzzling, but with a different focus that, I think, might appeal even to those for whom mathematics may not be their cup of tea. I call them numerologic puzzles—puzzles involving the use of logic in the area of numbers.

Modern-day interest in arithmetically-based puzzles can probably be traced to the invention of cryptarithms, a term coined by puzzlist Simon Vatriquant in the May, 1931 issue of the Belgian puzzle magazine, The Sphinx. Since then, cryptarithms and their variants have become favorites of math teachers throughout the world, since they stimulate mental processes that purportedly allow students to grasp the basic structure of the arithmetical operations.

A cryptarithm is a puzzle in which some or all of the digits in an addition, subtraction, division, multiplication, or other arithmetical layout have been deleted. You are asked to reconstruct the layout by deducing numerical values on the basis of the mathematical relationships indicated by the various arrangements and locations of the given numbers. Cryptarithms are, in effect, the arithmetical counterparts of cryptograms.

In a numerologic puzzle, the goal is not to reconstruct a layout, but rather to envision in a set of given numbers the arithmetical relations that allow you to group them together logically into an equation.

You are given certain numbers and asked to unite them with the relevant arithmetical signs into an equation. You must use all the numbers given. Here’s an example:

13, 75, 248, 4

Answer: 4(75 - 13) = 248

As you can see, these puzzles involve knowledge of principles of structure and construction, thus putting on display how logical thinking might unfold in microcosm. They also bring out parallels between language syntax (putting words together in sentences so that they make sense) and mathematical structures such as equations.

Following are 10 numerologic puzzles. They are not organized in any particular order of difficulty or complexity. In my view, they are truly challenging. (There may be more than one way to solve each puzzle and, indeed, I welcome readers to pass on to me any different solutions they might discover.)

(1)     23, 3, 63, 2

(2)     3, 2, 2, 2, 4

(3)     5, 5, 3, 5, 70

(4)     12, 2, 12, 38, 10

(5)     2915, 55, 55, 55, 55

(6)     6, 6, 6, 6, 215, 5

(7)     10, 0, 22, 12

(8)     1, 7, 8, 88, 88

(9)     2, 0, 100, 100, 100

(10)   1, 2, 3, 265, 9, 99

I know of no specific research connecting cryptarithms or similar types of puzzles to brain health, but my guess is that the kind of thinking involved likely does stimulate various parts of the brain to its benefit. Several processes seem to occur in tandem during the solution of the numerologic puzzles—sequencing of symbols logically, connecting them in a structurally-valid way, and inferring the relation of the parts to the whole.

Answers

[Note: there may be different ways or different signs of showing some of the operations, but each different layout should provide the same result]

(1)     3(23 - 2) = 63

(2)     23 - 22  = 4

(3)     3(5 x 5) - 5 = 70

(4)     2(12 + 12) - 10 = 38

(5)     (55 x 55) - (55 + 55) = 2915

(6)     6(6 x 6) - (6 - 5) = 215

(7)     (22 - 12) - 10 = 0

(8)     (88 - 7) - (88 - 8) = 1

(9)     (100 + 100) - 2(100) = 0

(10)   3(99 - 9 - 1) - 2 = 265