Arithmetic | justpuzzles

Each cell in this triangle is the sum of the two cells below it. Can you complete them?
1) triangle 1^*

				75
			43		.
		28		.		.
	18		.		.		.
.		.		.		.		8

I first encountered this type of puzzle in the Dutch translation of “One minute puzzles”, published by Arcturus Publishing Limited, London. This book published the numbers in circles, and all puzzles had the difficulty level of the one above, where there is always at least one cell which can be calculated with a simple addition or subtraction.
I replaced the circles with the pyramid pictured above, and in the following puzzles you will find an extra difficulty level introduced.

2) triangle 2^*

				.
			.		.
		18		.		15
	.		.		.		6
7		.		.		.		2

3) triangle 3^*

				115
			.		.
		19		.		36
	.		.		.		.
7		.		.		.		8

4) triangle 4^*

				80
			.		.
		18		.		24
	.		.		.		.
.		1		.		4		.

This type of puzzles exercises the parts of your brain which performs the arithmetic. If I interpret this article correctly, that is the horizontal segment of the bilateral intraparietal sulcus (HIPS), together with the precentral sulcus and inferior frontal gyrus.

You can check your solutions at solution 221, solution 231, solution 241, and solution 212.

In a previous post I mentioned inductive and deductive reasoning. One might say that in inductive reasoning all elements are known, while deductive logic tries to find the general underlying a pattern.

Famous game inventor Robert Abott designed a game “Eleusis” which is based on inductive reasoning, and though this is a blog on puzzles I hope to spend a future article on this game – there are sufficient similarities.

Another area where inductive reasoning is used, is in many IQ tests. Many of them have exercises consisting of a series of numbers, where the person taking the test is asked to find the next number. Patterns are usually purely based on elementary arithmetic.

Here are some exercises, ranging from elementary to difficult:
1) 3, 6, 9, …
2) 2, 6, 10, 14, …
3) 3, 12, 48, …
4) 19, 15, 11, …
5) 128, 64, 32, …
The sequences above are very simple, one operation is repeated again and again.

Things can however be made slightly more complicated, and this is the level you will find in many IQ tests.
6) 3, 6, 4, 8, 6, 12, 10, … (solution 41)
7) 3, 5, 8, 10, 13, 15, 18, .. (solution 51)
8 ) 3, 8, 3, 11, 3, 14, 3, .. (solution 7)
9) 100, 90, 180, 170, 340, 330, .. (solution 14)
10) 260, 130, 120, 60, 50, … (solution 20)
11) 15, 7, 22, 14, 29, 28, … (solution 28)
12) 18, 36, 13, 18, 8, 9, … (solution 35)
The nice thing about this type of puzzles is that after doing a series, you know the patterns to look for, which makes the next one easier to solve. As a result, you actually score higher when you consecutively take an IQ test where they have this kind of exercises. So yes, solving these puzzles actually makes you score higher at IQ tests.

13) 5, 11, 23, 47, … (solution 58)
14) 38, 22, 14, 10, … (solution 64)
15) 4, 12, 26, 54, … (solution 72)
16) 248, 86, 32, 14, … (solution 79)
17) 3, 4, 7, 11, … (solution 84)
18) 3, 6, 11, 18, … (solution 94)
19) 3, 7, 12, 18, 25, … (solution 104)
20) 3, 5, 9, 15, 25, … (solution 114)
21) 2, 5, 10, 17, 26, … (solution 124)
22) 3, 10, 29, … (solution 133)
23) 4, 8, 24, 96, 480, … (solution 142)

Occasionally, you will encounter sequences which will be posed in a different form, such as:
24) 5, 9, …, 17, …, …, 29. (solution 149)

25) One form occasionally used in puzzle books and magazines is the rectangle or square:

7	12	5
17	16	18
9	3	..

(solution 156)

26) Puzzle magazines often love to add illustrations. That is nothing new, Sam Loyd in the 19th century added an illustration to almost any puzzle in his Encyclopedia of Puzzles. Here is an example as they might appear in puzzle magazines.
In the supermarket, you find several fruit bags with labels. Unfortunately, one of the labels is damaged. Can you figure out what the price on the damaged label is?

Should the puzzle beat you, you can look up the solution 37

Things can be taken too far. For example, take the next sequence:
1, 2, 4, 8, 16, 31, ….
No, the 31 is correct, it is not a typo for 32, it is really 31.

You may spend days thinking about this sequences and never find an answer. Unless you happen to have stumbled on the problem before, you are not likely to stumble on the solution, which is 57.
The logic behind this series is that it is the number of regions formed by joining points on a circle:

Further reading:
http://www.nextnumber.com
http://www.nextnumber.com/show?39