Technique

Maximise one digit by minimising the others

Arithmetic • Number System • Digit Problems

Statement

If several digits have a fixed sum, one of them is largest when every other digit takes its smallest legal value, namely

0

for an ordinary place and

1

for a leading digit, provided the result is still at most

9

How to use it

1Reduce the problem to an equation on the digit sum first, then push the other digits down.
2Check the value obtained is at most $9$ ; if it exceeds $9$ the maximum is $9$ and the other digits absorb the remainder.
3For a minimum, reverse the move and push the other digits up to $9$ .

The digits of a three-digit number add up to

7

. What is the largest possible tens digit?

Step-by-step

1
$a + b + c = 7$ with $a \geq 1$ , $c \geq 0$
Why: The hundreds digit of a three-digit number cannot be $0$ .
2
$b = 7 - a - c$ is largest at $a = 1$ , $c = 0$
3
$b = 6$ , attained by $160$
Why: Exhibit a number that achieves the bound.

Answer

6

Watch out

The minimum chosen for the other digits must be legal: a leading digit stays at

1

, and a requirement that the digits be distinct or non-zero raises the floor.

Builds on

Also known as

largest possible digit, maximum value of a digit, minimise the other digits