Lose the Change

My Dearly Beloved, a professional philosopher, had explained to me once a fine point in the theory of just what it means to know something. I wouldn’t presume to try explaining that point (though I think I have it), but a core part of it is the thought experiment of remembering having put some change — we used a dime and a nickel — in your pocket, and finding later that you did have that same amount of money although not necessarily the same change — say, that you had three nickels instead.

That spun off a cute little side question that I’ll give to any needy recreational mathematician. It’s easy to imagine this problem where you remember having 15 cents in your pocket, and you do indeed have them, but you have a different number of coins from what you remember: three nickels instead of a dime and a nickel. Or you could remember having two coins, and indeed have two, but you have a different amount from what you remember: two dimes instead of a dime and a nickel.

Is it possible to remember correctly both the total number of coins you have, and the total value of those coins, while being mistaken about the number of each type? That is, could you remember rightly you have six coins and how much they add up to, but have the count of pennies, nickels, dimes, and quarters wrong? (In the United States there are also 50-cent and dollar coins minted, but they’re novelties and can be pretty much ignored. It’s all 1, 5, 10, and 25-cent pieces.) And can you prove it?