On the Subject of Missing Sequence

Hold up, these numbers aren’t consistent. Where’s the pattern?

Initially, the module contains a black, blank screen.

When pressed, the screen will show six rows of horizontally scrolling rectangles.

Each row contains six differently coloured rectangles. Most rectangles contain a number, but a few of them contain a question mark instead.

From top to bottom, the rectangles that share the same colour on each of the rows form a number sequence. Deduce and submit the number that should replace the question mark in the sequence for each colour to solve the module.

Sequence Info

The pattern in the sequence may not be obvious at first glance. Refer to the next page for some examples that may come up in the module.
The pattern usually relates with one (or more) of the following:

• Arithmetic and geometric offsets
• Prime numbers
• Recursive functions, e.g. Fibonacci sequence
• Perfect squares, perfect cubes
• Digital roots, Sum of digits, Product of digits
• Factors, mainly prime factors of numbers

Note that all the terms would always be integers.

Submission Info

Once you have all the answers ready, press the screen to enter submission mode. Note that the only way of getting back to the scrolling state is to strike on the module, so be careful!

In submission mode, the screen shows the six colored rectangles from the earlier state, but their values have been defaulted to 0.

The currently selected rectangle is denoted by the yellow highlight around it. Enter the value that belongs to the colour rectangle using the keypad on the right. Press the gray button to negate the value.

Note that all rectangles only accepts four digits. Leading zeros are always removed too.
To select a different rectangle, press the green button.

Press the screen again to submit the six values. If all of them are correct, the module will solve. Otherwise, the module will show the wrongly inputted rectangles in red. Press the screen to change it back to its scrolling state. The module doesn’t reset.

After a strike, the module only requires you to input incorrect values. This means the correct submitted rectangles will not be selected.

Example Sequence (1)

Given sequence: 102, 96, 108, 84, 132, ??

By taking their arithmetic offsets the pattern would be clear:

Arithmetic offsets: -6, 12, -24, 48, ??

Thus it’s clear that the offsets form a geometric sequence with offset -2, starting from -6. Therefore the next offset should be -96, which yields 36 as our answer.

Example Sequence (2)

Given sequence: 4, 10, 30, ??, 420, 1890

By taking their geometric offsets the pattern would be clear too:

Geometric offsets: 2.5, 3, ??, ??, 4.5

It’s not hard to realise the offsets are an arithmetic sequence with offset 0.5 starting from 2.5, so the question marks are basically 30×3.5=105.

Example Sequence (3)

Given sequence: ??, 124, 213, 337, 550, 887

By observing the sequence, the pattern would be clear:

Observing: ??, 124, 213, 124+213=337, 213+337=550, 337+550=887

Hence, the sequence uses the Fibonacci progression, i.e. the next term is the sum of the previous two terms. A bit of backtracking gives 89 as the answer.

Example Sequence (4)

Given sequence: ??, 29, 127, 345, 731, 1333

With a bit of observing, it’s clear that the terms are perfect cubes plus 2, as follows:

Observing: ??, 3^3+2=29, 5^3+2=127, 7^3+2=345, 9^3+2=731, 11^3+2=1333

Therefore the answer is clearly 1^3+2=3.

Example Sequence (5)

Given sequence: 168, 183, 195, 210, ??, 219

Once again, let’s take their arithmetic offsets to find out the pattern:

Arithmetic offsets: 15, 12, 15, ??, ??

At first glance, the sequence might be just alternating 15 and 12, but it’s quickly disproved since the last term is not 225. However, by looking closely, the offsets are actually the sum of each digit of the previous terms, which is in fact true:

Observing: 1+6+8=15, 1+8+3=12, 1+9+5=15, 2+1+0=3, ??

The sum of digits of 210 is 3, so add that to 210 would be the answer 213, which is also proven true since 213 + (2+1+3) = 213 + 6 = 219.

Example Sequence (6)

Given sequence: 2, 7, ??, 52, 128, 304

This sequence is a bit hard to realise, but by factorizing and clever grouping then the pattern would be obvious:

Factorizing: 2, 7, ??, 2^2×13, 2^7, 2^4×19

Grouping: 0.5×4, 1×7, ??, 4×13, 8×16, 16×19

It would be clear that in this sequence, the left numbers form a geometric sequence with offset 2, starting from 0.5 (or 1/2), and the right numbers form an arithmetic sequence with offset 3, starting from 4.

Therefore the answer should be 2×10 = 20.