Number Sequence Puzzles
Below may be found information on, and quizzes of, number sequences puzzles. Click on a link below to go to the section you want to browse.
Some information on number sequences
Number sequence puzzles — test 1 (easy)
Number sequence puzzles — test 2 (medium)
Number sequence puzzles — test 3 (hard)
Some information on number sequences
The numbers in a number sequence are called terms.
The n-th term in a sequence can be denoted by un .
If sequence A contains the same terms as sequence B, in the same order, but possibly with some terms missing, then sequence A is said to be a subsequence of sequence B.
If a term has position n in a sequence, then its index (plural: indices) is n. Thus, the first term has index 1, the second term has index 2, the third term has index 3, and so on.
Those terms with odd index are called odd terms. Thus, if a sequence A has n-th term un , then the odd terms comprise the following subsequence:
u1 , u3 , u5 , u7 , u9 , …
Those terms with even index are called even terms. Thus, if a sequence A has n-th term un , then the even terms comprise the following subsequence:
u2 , u4 , u6 , u8 , u10 , …
In a number sequence puzzle, you are given the beginning of a number sequence and then you have to work out what the next one or two terms are likely to be, under the assumption that there is some simple rule that connects the terms. For example, with the sequence
1, 4, 7, 10, 13, 16, …
it is apparent that each term after the first is equal to the previous term plus three. Hence the next term in the sequence will be
16 + 3 = 19.
Here is another example. For the sequence
3, 12, 48, 192, 768, 3072, …
we see, after brief reflection, that, after the first term, each term is four times the previous term. Hence the next term in the sequence will be
3072 x 4 = 12288.
There are many possible rules for number sequence puzzles, as you will discover if you try the tests below.
Number sequence puzzles — test 1 (easy)
Ten examples of number sequence puzzles are given below.
For each puzzle, the clickable solution states both the answer and the rule used to obtain it.
Question 1:
For the sequence below, find the next number and the rule used to obtain it.
2, 4, 6, 8, 10, ?
Click to reveal solution
Answer: 12.
The rule: Even numbers.
Question 2:
For the sequence below, find the next number and the rule used to obtain it.
16, 10, 4, -2, -8, ?
Click to reveal solution
Answer: -14.
The rule: Subtract 6 from the previous term.
Question 3:
For the sequence below, find the next number and the rule used to obtain it.
23, 41, 59, 77, 95, ?
Click to reveal solution
Answer: 113.
The rule: Add 18 to the previous term.
Question 4:
For the sequence below, find the next number and the rule used to obtain it.
3, 6, 12, 24, 48, ?
Click to reveal solution
Answer: 96.
The rule: Multiply the previous term by 2.
Question 5:
For the sequence below, find the next two numbers and the rule used to obtain them.
5, 6, 5, 6, 5, 6, ?, ?
Click to reveal solution
Answer: 5, 6.
The rule: Repetition of the numbers 5 and 6.
Question 6:
For the sequence below, find the next number and the rule used to obtain it.
448, 224, 112, 56, 28, ?
Click to reveal solution
Answer: 14.
The rule: Divide the previous term by 2.
Question 7:
For the sequence below, find the next number and the rule used to obtain it.
5, 11, 23, 47, 95, ?
Click to reveal solution
Answer: 191.
The rule: Double the previous term and add 1.
Question 8:
For the sequence below, find the next number and the rule used to obtain it.
1, 4, 9, 16, 25, ?
Click to reveal solution
Answer: 36.
The rule: The square numbers.
Question 9:
For the sequence below, find the next number and the rule used to obtain it.
654, 645, 564, 546, 465, ?
Click to reveal solution
Answer: 456.
The rule: Those numbers, written in decreasing order, that contain the three digits 4, 5, and 6.
Question 10:
For the sequence below, find the next number and the rule used to obtain it.
200, 827, 1454, 2081, 2708, 3335, ?
Click to reveal solution
Answer: 3962.
The rule: Add 627 to the previous term.
Number sequence puzzles — test 2 (medium)
Ten examples of number sequence puzzles are given below.
For each puzzle, the clickable solution states both the answer and the rule used to obtain it.
Question 1:
For the sequence below, find the next number and the rule used to obtain it.
3, −15, 75, −375, 1875, −9375, ?
Click to reveal solution
Answer: 46875.
The rule: Multiply the previous term by -5.
Question 2:
For the sequence below, find the next number and the rule used to obtain it.
2, 3, 5, 7, 11, ?
Click to reveal solution
Answer: 13.
The rule: The prime numbers.
Question 3:
For the sequence below, find the next number and the rule used to obtain it.
1, 2, 4, 8, 16, ?
Click to reveal solution
Answer: 32.
The rule: Powers of 2.
Question 4:
For the sequence below, find the next number and the rule used to obtain it.
−32, 48, −72, 108, −162, ?
Click to reveal solution
Answer: 243.
The rule: Multiply the previous term by -3/2.
Question 5:
For the sequence below, find the next number and the rule used to obtain it.
1, 3, 6, 10, 15, 21, ?
Click to reveal solution
Answer: 28.
The rule: The triangular numbers.
Question 6:
For the sequence below, find the next two numbers and the rule used to obtain them.
1, 1, 2, 3, 5, 8, 13, 21, ?, ?
Click to reveal solution
Answer: 34, 55.
The rule: The Fibonacci numbers — each term, from the third term onwards, is the sum of the two previous terms.
Question 7:
For the sequence below, find the next number and the rule used to obtain it.
10, 36, 114, 348, 1050, 3156, ?
Click to reveal solution
Answer: 9474.
The rule: Triple the previous term and add 6.
Question 8:
For the sequence below, find the next three numbers and the rule used to obtain them.
1, 6, 7, 1, 6, 7, 1, 6, 7, ?, ?, ?
Click to reveal solution
Answer: 1, 6, 7.
The rule: Repetition of the numbers 1, 6, 7.
Question 9:
For the sequence below, find the next number and the rule used to obtain it.
2, 3, 8, 63, 3968, ?
Click to reveal solution
Answer: 15745023.
The rule: Square the previous term and subtract 1.
Question 10:
For the sequence below, find the next number and the rule used to obtain it.
1, 8, 27, 64, 125, ?
Click to reveal solution
Answer: 216.
The rule: The cubes.
Number sequence puzzles — test 3 (hard)
Ten examples of number sequence puzzles are given below.
For each puzzle, the clickable solution states both the answer and the rule used to obtain it.
Question 1:
For the sequence below, find the next number and the rule used to obtain it.
3, 1, 4, 1, 5, ?
Click to reveal solution
Answer: 9.
The rule: The decimal digits of the number π = 3.14159…
Question 2:
For the sequence below, find the next two numbers and the rule used to obtain them.
1, 2, 4, 3, 9, 5, 16, 7, ?, ?
Click to reveal solution
Answer: 25, 11.
The rule: The odd terms are the squares, and the even terms are the primes.
Question 3:
For the sequence below, find the next number and the rule used to obtain it.
15, 18, 1, 14, 7, ? Hint: Colour
Click to reveal solution
Answer: 5.
The rule: The positions in the alphabet of the letters in ORANGE.
Question 4:
For the sequence below, find the next three numbers and the rule used to obtain them.
7, 2, 3, 12, 17, 32, 61, 110, 203, ?, ?, ?
Click to reveal solution
Answer: 374, 687, 1264.
The rule: Each term, from the fourth term onwards, is the sum of the three previous terms.
Question 5:
For the sequence below, find the next number and the rule used to obtain it.
557, 560, 569, 596, 677, 920, ?
Click to reveal solution
Answer: 1649.
The rule: un=un-1+3n-1 for n≥2, with initial condition (first term) u1=557.
Question 6:
For the sequence below, find the next number and the rule used to obtain it.
7, 9, 13, 21, 37, ?
Click to reveal solution
Answer: 69.
The rule: un=2n+5 for n≥1.
Question 7:
For the sequence below, find the next number and the rule used to obtain it.
85085, 5, 7, 11, 13, ?
Click to reveal solution
Answer: 17.
The rule: The first term is a large product of primes, and the remaining terms are its prime factors in increasing numerical order.
Question 8:
For the sequence below, find the next number and the rule used to obtain it.
6, 28, 496, 8128, 33550336, 8589869056, ?
Click to reveal solution
Answer: 137438691328.
The rule: The perfect numbers.
Question 9:
For the sequence below, find the next number and the rule used to obtain it.
−361, 125, −37, 17, −1, 5, ?
Click to reveal solution
Answer: 3.
The rule: Subtract 14 from the previous term, and multiply by -1/3.
Question 10:
For the sequence below, find the next number and the rule used to obtain it.
16, 36, 64, 81, 100, 144, ?
Click to reveal solution
Answer: 196.
The rule: The squares of the composite numbers.