Solution for Largest Bet

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The answer is $27. Here's how I got it.

Start with 2 simple equations:

(I'm using "!=" to mean "not equal".)

1. A8 + B5 != N
2. N = CD

In these equations "A", "B" and "C" are the value of the ten's digit, and "D" is the value of the one's digit.

"N" is the largest bet that can't be made.

If I can find a value for "C" which makes equation 1 false for the values "C1", "C2", "C3" and "C4" ("C0" and "C5" will always be false as they will always be a multiple of 5)

then "N" must be < "C1" because:

• C6 = C1 + 5
• C7 = C2 + 5
• C8 = C3 + 5
• C9 = C4 + 5
• and any higher ten's digit is a multiple of 5 (bet can be placed with additional $5 chips)

Since we're looking for the smallest value of "C", I'll start with 1:

For C = 1
A8 + B5 != 11	True	(no combination of 8 and 5 equals 11)
A8 + B5 != 12	True	(no combination of 8 and 5 equals 12)
A8 + B5 != 13	True	(no combination of 8 and 5 equals 13)
A8 + B5 != 14	True	(no combination of 8 and 5 equals 14)

Now try 2:

For C = 2
A8 + B5 != 21	False	8 + 8 + 5 = 21
A8 + B5 != 22	True	no combination of 8 and 5 equals 22
A8 + B5 != 23	False	8 + 5 + 5 + 5 = 23
A8 + B5 != 24	False	8 + 8 + 8 = 24

Now try 3:

For C = 3
A8 + B5 != 31	False	8 + 8 + 5 + 5 + 5 = 31
A8 + B5 != 32	False	8 + 8 + 8 + 8 = 32
A8 + B5 != 33	False	8 + 5 + 5 + 5 + 5 + 5= 33
A8 + B5 != 34	False	8 + 8 + 8 + 5 + 5 = 34

So the largest bet that can't be made must be < 31. Now just test each case until you get the answer:

30	wrong because	5 + 5 + 5 + 5 + 5 + 5 = 30
29	wrong because	8 + 8 + 8 + 5 = 29
28	wrong because	8 + 5 + 5 + 5 + 5 = 28
27	right because	no combination of 8 and 5 equals 27

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