Generating valid currency by cutting notes

Question

Consider the following problem:

• A $\$50$ is still considered valid currency if you have $70\%$ of the full note.
• The $70\%$ of the full note can consist of at-most two pieces.
• The two pieces are glued together, and agree on their boundary.
• Let us assume that each complete note is $100\%$ identical. (To avoid identification codes etc).

Can we generate new valid currency by cutting up full $\$50$ notes? If not, what is the largest percentage of the full note (as opposed to $70\%$) that would allow us to generate new currency.

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Evidently if $70\%$ was changed to $50\%$, we could just cut any note into halves, and obtain two valid notes.

Side note: I don't plan to deface currency, which is illegal, this is just a curiosity, that I couldn't solve as of yet.

Answer 1

You can cut $7$ notes as shown below, and glue the pieces into $10$ new $(7/10)$ notes. Each note can be thought of as a"hand" containing $3/10$ of the area, attached to seven "fingers" with $1/10$ each (not all fingers will be separated by cuts). The numbers and colours indicate which finished note each piece goes to.

Thus the finished notes consist of:

1. hand and top 4 fingers of note 1.
2. bottom 3 fingers of note 1 and top 4 fingers of note 2.
3. hand and bottom 3 fingers of note 2, top finger of note 3.
4. hand and fingers 2 to 4 of note 3.
5. bottom 2 fingers of note 3, top 5 fingers of note 4.
6. hand and bottom 2 fingers of note 4, fingers 4 and 5 of note 5.
7. hand and fingers 2,3,6,7 of note 5.
8. top 1 finger of note 5, bottom 6 fingers of note 6.
9. hand and top finger of note 6, fingers 2,3,4 of note 7.
10. hand and fingers 1, 5,6,7 of note 7.