This question's explanation and your answers so far.
“甘味”，“うま味”，“塩味”，“酸味”，“苦味”の5種類の味覚を，6ビット（2進数で6けた）の数値で符号化する。これらを組み合わせた複合味を，数値の加減算で表現できるようにしたい。例えば，“甘味”と“酸味”を組み合わせた“甘酸っぱい”という複合味の符号を，それぞれの数値を加算して表現するとともに，逆に“甘酸っぱい”から“甘味”成分を取り除いた“酸味”を減算で表現できるようにしたい。味覚の符号として，適切なものはどれか。
| 甘味 | うま味 | 塩味 | 酸味 | 苦味 | |
|---|---|---|---|---|---|
| ア | 000000 | 000001 | 000010 | 000011 | 000100 |
| 甘味 | うま味 | 塩味 | 酸味 | 苦味 | |
|---|---|---|---|---|---|
| イ | 000001 | 000010 | 000011 | 000100 | 000101 |
| 甘味 | うま味 | 塩味 | 酸味 | 苦味 | |
|---|---|---|---|---|---|
| ウ | 000001 | 000010 | 000100 | 001000 | 010000 |
| 甘味 | うま味 | 塩味 | 酸味 | 苦味 | |
|---|---|---|---|---|---|
| エ | 000001 | 000011 | 000111 | 001111 | 011111 |
Since sweetness is 000000 (the value 0), adding sweetness to any taste leaves the code unchanged, so “sweet-and-sour” and “sourness” both become the same 000011 and cannot be told apart. In addition, umami 000001 + saltiness 000010 = 000011 matches the code for sourness, so a compound taste collides with a different single taste.
The codes run as the consecutive numbers 1, 2, 3, 4, 5, so sweetness 000001 + umami 000010 = 000011 matches the code for saltiness, and sweetness 000001 + sourness 000100 = 000101 matches the code for bitterness. The sum for a compound taste equals the value of a single taste, so the contents cannot be identified uniquely from the code.
正解 Option U assigns sweetness=000001, umami=000010, saltiness=000100, sourness=001000, and bitterness=010000, giving each of the five tastes its own independent bit that does not overlap with any other. Because the bit positions never overlap, addition produces no carry, so the bit pattern of the sum directly represents the set of tastes that are included. Following the example in the question, sweet-and-sour = sweetness 000001 + sourness 001000 = 001001, and the two bits set to 1 indicate sweetness and sourness. Removing the sweetness component is likewise expressed correctly by subtraction: 001001 − 000001 = 001000 = sourness. Because the sum itself represents the set of components, and compound tastes can be composed and decomposed by addition and subtraction, U is the correct encoding.
In the codes 1, 3, 7, 15, 31 the lower bits overlap one another, so addition produces carries and the bit string of the sum is no longer a superposition of the component bits. For example, sweetness 000001 + sourness 001111 = 010000, and the sum retains neither the bits of sweetness nor those of sourness, so you cannot read off from the code which tastes are included. It does not make each single bit correspond to one taste the way U does, so it is inappropriate.