Solving 'The Change-making problem' with a formula

Patrick2788 Good stuff. There was a video shared by Diarmuid Early on YouTube a few months ago called "Recursive LAMBDAs to traverse a tree", where he walked through solving the Peg Game Excel challenge. Seems like the same logic could be applied here... in any case, you might find it interesting.

Just for fun, here's another "Greedy" LAMBDA function based on my previous reply, but with additional options that can output 4 different layouts. The generic syntax is:

=MAKECHANGE(amount, denoms, [images], [list_mode])

• amount - The amount due
• denoms - The range containing the denomination values
• images - [optional] The range containing the denomination images (if omitted, the function will return the denomination values by default)
• list_mode - [optional] 0 or FALSE = summary report grouped by denomination (default); 1 or TRUE = complete list of all denominations

This function is highly efficient and can handle amounts greater that 1 trillion with ease (summary report). When list_mode is set to TRUE, it can only handle amounts up to 100 billion, due to the row limitations of TOCOL. The complete definition is:

MAKECHANGE:
=LAMBDA(amount,denoms,[images],[list_mode],
    LET(
        d, SORT(denoms,, -1),
        m, MAX((d<=amount)*d),
        r, IF(m=0, 0, REDUCE(HSTACK(m, INT(amount/m)), d, LAMBDA(v,n,
            LET(a, ROUND(amount-SUMPRODUCT(TAKE(v,, 1)*TAKE(v,,-1)), 2),
                IF(a<n, v, VSTACK(v, HSTACK(n, INT(a/n)))))))),
        x, TAKE(r,, 1),
        y, TAKE(r,,-1),
        t, SUMPRODUCT(x*y),
        h, EXPAND(HSTACK("Amount Tendered:", t, IF(ROUND(amount-t, 2), "⮽", "☑")), 2,, ""),
        IF(
            list_mode,
            LET(z, IFERROR(TOCOL(IFS(y>=SEQUENCE(, MAX(y)), x), 2), 0), IFNA(VSTACK(h, "Denominations:",
                WRAPROWS(IF(ISOMITTED(images), z, XLOOKUP(z, denoms, images, "")), 4)), "")),
            VSTACK(h, HSTACK("Denominations", "Quantity", "Amount"),
                HSTACK(IF(ISOMITTED(images), x, XLOOKUP(x, denoms, images, "")), y, x*y))
        )
    )
)

Enjoy!