# 112

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## Number

$112$ (one hundred and twelve) is:

$2^4 \times 7$

The $7$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$:
$112 = 1 + 7 + 11 + 16 + 21 + 26 + 31 = \dfrac {7 \left({5 \times 7 - 3}\right)} 2$

The $30$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $72$, $76$, $92$, $96$, $108$, $112$, $\ldots$

The $53$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $95$, $96$, $100$, $101$, $102$, $107$ which cannot be expressed as the sum of distinct pentagonal numbers

The $16$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$:
$112 = 56 \times 2 = 56 \times \left({1 \times 1 \times 2}\right)$

The length of the side of the smallest perfect square dissection of an integer square

The side length of the smallest equilateral triangle with sides of integer length which contains a point which is an integer distance from each vertex

## Historical Note

$112$ is the number of pounds avoirdupois in a hundredweight.