Maximum count

Also called: largest count, highest count, max count

The largest value a counter reaches before wrapping. An n-bit binary counter's maximum count is 2ⁿ − 1, one less than its modulus.

The maximum count is the highest number a counter shows. For a counter that starts at 0 it's always one less than the modulus, because 0 takes up one of the states.

An n-bit binary counter: 2ⁿ − 1. With 8 bits that's 255; with 16 bits, 65 535. It's the same as the largest unsigned number with n bits.
A mod n counter: N − 1. A BCD counter's maximum is 9.

The question often runs the other way: how many flip-flops to count up to M? There are M + 1 states (0 to M), so you need the smallest n with 2ⁿ ≥ M + 1 (see flip flops needed).

The terminal count output marks the maximum count in an up counter, because that's the state just before it wraps.

Worked examples

Example

From flip-flops to maximum count

What is the maximum count of a 6-bit binary up counter?

1.
2⁶ = 64 states.
2.
They run 0 to 63.
3.
Maximum count = 63 = 111111.

Example

From maximum count to flip-flops

A counter must count from 0 up to 200. How many flip-flops?

1.
0 to 200 inclusive is 201 states.
2.
2⁷ = 128 is too few; 2⁸ = 256 is enough.
3.
So 8 flip-flops, with 255 − 200 = 55 codes unused.

Common mistakes

Using the maximum count as the number of states. Counting to 200 needs 201 states.
Saying an n-bit counter counts to 2ⁿ. It counts to 2ⁿ − 1.

Practice Maximum count

Interactive questions with instant feedback and a worked solution for every wrong answer.

Counters lesson full course

What is the largest value, in decimal, that a 6-bit up counter can show?

Practice exams

Counters: Exam 2 · 1 question

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