Mathematicshard

James Is a Famous Spy and a Math Genius - Notebook of 100 Pages

Question

James is a famous spy and a math genius. He is spying on an enemy base and has lost all his tools, having only a notebook of 100 pages. He can fold the notebook to observe certain page combinations. How many unique observations can he make?

A combinatorics puzzle about page pairs visible through notebook folds.

Unique Observations

99 unique fold positions, each revealing a distinct page pairing

1Understand the Setup

James has a 100-page notebook (pages 1–100). When the notebook is closed, pages are stacked in order. Folding the notebook at a specific point exposes page pairs that are not normally adjacent.

  Notebook Structure (side view):

  Page 1  │  Page 2  │  Page 3  │ ... │ Page 100
  ────────┼─────────┼─────────┼─────┼─────────

  Fold at page k → pages k and k+1 meet

2Model the Folding

When you fold the notebook at page

k

, you bring page

k

to meet page

k+1

. The visible pages after folding depend on where you fold. Each fold exposes a pair of pages that aren't normally adjacent.

Folding Mechanics

Folding at position

k

(between pages

k

and

k+1

) mirrors the pages on the left side onto the right side. Pages symmetric around the fold point become visible pairs.

3Count the Fold Points

A 100-page notebook can be folded between any two consecutive pages. There are **99 possible fold points**: between pages 1–2, pages 2–3, ..., pages 99–100.

Fold Position	Left Page	Right Page	Observation
1	Page 1	Page 2	Pages 1-2 face each other
2	Page 2	Page 3	Pages 2-3 visible
...	...	...	...
50	Page 50	Page 51	Mid-point fold
...	...	...	...
99	Page 99	Page 100	Last fold

4Determine Unique Observations

Each fold point

k

(for

k = 1, 2, \ldots, 99

) creates a distinct observation because the page pairings change with each position. No two fold points produce the exact same set of visible page pairs.

Total pages100

Possible fold points99

Each fold creates unique pairingsYes

UNIQUE OBSERVATIONS99

5Key Concepts

Combinatorial Thinking

The key insight is that each of the 99 inter-page boundaries defines a unique fold, and each fold produces a unique observation. No additional complexity from double-folds or partial folds is needed.

Symmetry in Folding

When folded at position

k

, page

j

(where

j \leq k

) aligns with page

2k - j + 1

(if that page exists). This mirror-image relationship means each fold is geometrically distinct.

Quiz

Test your understanding with these questions.

How many possible fold points exist in a 100-page notebook?

When James folds the notebook, what does each fold reveal?